import Mathlib import Mathlib.Analysis.SpecialFunctions.Pow.Real set_option linter.unusedVariables.analyzeTactics true open Real lemma imo_2023_p4_1 (x a: ℕ → ℝ) (hxp: ∀ (i : ℕ), 0 < x i) (h₀: ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) : ∀ (n : ℕ), (1 ≤ n ∧ n ≤ 2022) → a (n) < a (n + 1) := by intros n hn have h₂: a n = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by refine h₀ n ?_ constructor . exact hn.1 . linarith have h₃: a (n + 1) = sqrt ((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k) := by refine h₀ (n + 1) ?_ constructor . linarith . linarith rw [h₂,h₃] refine sqrt_lt_sqrt ?_ ?_ . refine le_of_lt ?_ refine mul_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . exact fun i _ => hxp i . simp linarith . refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith . have g₀: 1 ≤ n + 1 := by linarith rw [Finset.sum_Ico_succ_top g₀ _, Finset.sum_Ico_succ_top g₀ _] repeat rw [add_mul, mul_add] have h₄: 0 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + x (n + 1) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + 1 / x (n + 1)) := by refine add_pos ?_ ?_ . refine mul_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . exact fun i _ => hxp i . simp linarith . exact one_div_pos.mpr (hxp (n + 1)) . refine mul_pos ?_ ?_ . exact hxp (n + 1) . refine add_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith . exact one_div_pos.mpr (hxp (n + 1)) linarith lemma imo_2023_p4_1_1 (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2022) : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by refine h₀ n ?_ constructor . exact hn.1 . linarith lemma imo_2023_p4_1_2 -- (x a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * -- Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2022) : 1 ≤ n ∧ n ≤ 2023 := by constructor . exact hn.1 . linarith lemma imo_2023_p4_1_3 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2022) (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) : a n < a (n + 1) := by have h₃: a (n + 1) = sqrt ((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k) := by refine h₀ (n + 1) ?_ constructor . linarith . linarith rw [h₂,h₃] refine sqrt_lt_sqrt ?_ ?_ . refine le_of_lt ?_ refine mul_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . exact fun i _ => hxp i . simp linarith . refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith . have g₀: 1 ≤ n + 1 := by linarith rw [Finset.sum_Ico_succ_top g₀ _, Finset.sum_Ico_succ_top g₀ _] repeat rw [add_mul, mul_add] have h₄: 0 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + x (n + 1) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + 1 / x (n + 1)) := by refine add_pos ?_ ?_ . refine mul_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . exact fun i _ => hxp i . simp linarith . exact one_div_pos.mpr (hxp (n + 1)) . refine mul_pos ?_ ?_ . exact hxp (n + 1) . refine add_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith . exact one_div_pos.mpr (hxp (n + 1)) linarith lemma imo_2023_p4_1_4 (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2022) : -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k) := by refine h₀ (n + 1) ?_ constructor . linarith . linarith lemma imo_2023_p4_1_5 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2022) (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) : a n < a (n + 1) := by rw [h₂,h₃] refine sqrt_lt_sqrt ?_ ?_ . refine le_of_lt ?_ refine mul_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . exact fun i _ => hxp i . simp linarith . refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith . have g₀: 1 ≤ n + 1 := by linarith rw [Finset.sum_Ico_succ_top g₀ _, Finset.sum_Ico_succ_top g₀ _] repeat rw [add_mul, mul_add] have h₄: 0 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + x (n + 1) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + 1 / x (n + 1)) := by refine add_pos ?_ ?_ . refine mul_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . exact fun i _ => hxp i . simp linarith . exact one_div_pos.mpr (hxp (n + 1)) . refine mul_pos ?_ ?_ . exact hxp (n + 1) . refine add_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith . exact one_div_pos.mpr (hxp (n + 1)) linarith lemma imo_2023_p4_1_6 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2022) : -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) : √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) < √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k) := by refine sqrt_lt_sqrt ?_ ?_ . refine le_of_lt ?_ refine mul_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . exact fun i _ => hxp i . simp linarith . refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith . have g₀: 1 ≤ n + 1 := by linarith rw [Finset.sum_Ico_succ_top g₀ _, Finset.sum_Ico_succ_top g₀ _] repeat rw [add_mul, mul_add] have h₄: 0 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + x (n + 1) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + 1 / x (n + 1)) := by refine add_pos ?_ ?_ . refine mul_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . exact fun i _ => hxp i . simp linarith . exact one_div_pos.mpr (hxp (n + 1)) . refine mul_pos ?_ ?_ . exact hxp (n + 1) . refine add_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith . exact one_div_pos.mpr (hxp (n + 1)) linarith lemma imo_2023_p4_1_7 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2022) : -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k := by refine le_of_lt ?_ refine mul_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . exact fun i _ => hxp i . simp linarith . refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith lemma imo_2023_p4_1_8 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2022) : -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) : 0 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k := by refine mul_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . exact fun i _ => hxp i . simp linarith . refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith lemma imo_2023_p4_1_9 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2022) : -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) : 0 < Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k := by refine Finset.sum_pos ?_ ?_ . exact fun i _ => hxp i . simp linarith lemma imo_2023_p4_1_10 -- (x : ℕ → ℝ) -- (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2022) : -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) : (Finset.Ico 1 (n + 1)).Nonempty := by simp linarith lemma imo_2023_p4_1_11 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2022) : -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) : 0 < Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k := by refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith lemma imo_2023_p4_1_12 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) : -- (hn : 1 ≤ n ∧ n ≤ 2022) -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) : ∀ i ∈ Finset.Ico 1 (n + 1), 0 < 1 / x i := by intros i _ exact one_div_pos.mpr (hxp i) lemma imo_2023_p4_1_13 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2022) : -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) : ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) < (Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k := by have g₀: 1 ≤ n + 1 := by linarith rw [Finset.sum_Ico_succ_top g₀ _, Finset.sum_Ico_succ_top g₀ _] repeat rw [add_mul, mul_add] have h₄: 0 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + x (n + 1) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + 1 / x (n + 1)) := by refine add_pos ?_ ?_ . refine mul_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . exact fun i _ => hxp i . simp linarith . exact one_div_pos.mpr (hxp (n + 1)) . refine mul_pos ?_ ?_ . exact hxp (n + 1) . refine add_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith . exact one_div_pos.mpr (hxp (n + 1)) linarith lemma imo_2023_p4_1_14 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2022) -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) (g₀ : 1 ≤ n + 1) : ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) < (Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k := by rw [Finset.sum_Ico_succ_top g₀ _, Finset.sum_Ico_succ_top g₀ _] repeat rw [add_mul, mul_add] have h₄: 0 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + x (n + 1) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + 1 / x (n + 1)) := by refine add_pos ?_ ?_ . refine mul_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . exact fun i _ => hxp i . simp linarith . exact one_div_pos.mpr (hxp (n + 1)) . refine mul_pos ?_ ?_ . exact hxp (n + 1) . refine add_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith . exact one_div_pos.mpr (hxp (n + 1)) linarith lemma imo_2023_p4_1_15 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2022) : -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) -- (g₀ : 1 ≤ n + 1) : ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) < ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + x (n + 1)) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + 1 / x (n + 1)) := by repeat rw [add_mul, mul_add] have h₄: 0 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + x (n + 1) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + 1 / x (n + 1)) := by refine add_pos ?_ ?_ . refine mul_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . exact fun i _ => hxp i . simp linarith . exact one_div_pos.mpr (hxp (n + 1)) . refine mul_pos ?_ ?_ . exact hxp (n + 1) . refine add_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith . exact one_div_pos.mpr (hxp (n + 1)) linarith lemma imo_2023_p4_1_16 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2022) : -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) -- (g₀ : 1 ≤ n + 1) : ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) < ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + x (n + 1) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + 1 / x (n + 1)) := by have h₄: 0 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + x (n + 1) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + 1 / x (n + 1)) := by refine add_pos ?_ ?_ . refine mul_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . exact fun i _ => hxp i . simp linarith . exact one_div_pos.mpr (hxp (n + 1)) . refine mul_pos ?_ ?_ . exact hxp (n + 1) . refine add_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith . exact one_div_pos.mpr (hxp (n + 1)) linarith lemma imo_2023_p4_1_17 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2022) : -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) -- (g₀ : 1 ≤ n + 1) : 0 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + x (n + 1) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + 1 / x (n + 1)) := by refine add_pos ?_ ?_ . refine mul_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . exact fun i _ => hxp i . simp linarith . exact one_div_pos.mpr (hxp (n + 1)) . refine mul_pos ?_ ?_ . exact hxp (n + 1) . refine add_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith . exact one_div_pos.mpr (hxp (n + 1)) lemma imo_2023_p4_1_18 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2022) : -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) -- (g₀ : 1 ≤ n + 1) : 0 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) := by refine mul_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . exact fun i _ => hxp i . simp linarith . exact one_div_pos.mpr (hxp (n + 1)) lemma imo_2023_p4_1_19 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2022) : -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) -- (g₀ : 1 ≤ n + 1) : 0 < Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k := by refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith lemma imo_2023_p4_1_20 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) : -- (hn : 1 ≤ n ∧ n ≤ 2022) -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) -- (g₀ : 1 ≤ n + 1) : ∀ i ∈ Finset.Ico 1 (n + 1), 0 < 1 / x i := by intros i _ exact one_div_pos.mpr (hxp i) lemma imo_2023_p4_1_21 -- (x : ℕ → ℝ) -- (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2022) : -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) -- (g₀ : 1 ≤ n + 1) : (Finset.Ico 1 (n + 1)).Nonempty := by simp linarith lemma imo_2023_p4_2 -- my_amgm (b1 b2 b3 b4 :ℝ) (hb1: 0 ≤ b1) (hb2: 0 ≤ b2) (hb3: 0 ≤ b3) (hb4: 0 ≤ b4) : (4 * (b1 * b2 * b3 * b4) ^ (4:ℝ)⁻¹ ≤ b1 + b2 + b3 + b4) := by let w1 : ℝ := (4:ℝ)⁻¹ let w2 : ℝ := w1 let w3 : ℝ := w2 let w4 : ℝ := w3 rw [mul_comm] refine mul_le_of_le_div₀ ?_ (by norm_num) ?_ . refine add_nonneg ?_ hb4 refine add_nonneg ?_ hb3 exact add_nonneg hb1 hb2 . have h₀: (b1^w1 * b2^w2 * b3^w3 * b4^w4) ≤ w1 * b1 + w2 * b2 + w3 * b3 + w4 * b4 := by refine geom_mean_le_arith_mean4_weighted (by norm_num) ?_ ?_ ?_ hb1 hb2 hb3 hb4 ?_ . norm_num . norm_num . norm_num . norm_num repeat rw [mul_rpow _] . ring_nf at * linarith repeat { assumption } . exact mul_nonneg hb1 hb2 . exact hb4 . refine mul_nonneg ?_ hb3 exact mul_nonneg hb1 hb2 lemma imo_2023_p4_2_1 (b1 b2 b3 b4 : ℝ) (w1 w2 w3 w4 : ℝ) (hb1 : 0 ≤ b1) (hb2 : 0 ≤ b2) (hb3 : 0 ≤ b3) (hb4 : 0 ≤ b4) (hw1 : w1 = (4:ℝ)⁻¹) (hw2 : w2 = w1) (hw3 : w3 = w1) (hw4 : w4 = w1) : (b1 * b2 * b3 * b4) ^ (4:ℝ)⁻¹ * (4:ℝ) ≤ b1 + b2 + b3 + b4 := by refine mul_le_of_le_div₀ ?_ (by norm_num) ?_ . refine add_nonneg ?_ hb4 refine add_nonneg ?_ hb3 exact add_nonneg hb1 hb2 . have h₀: (b1^w1 * b2^w2 * b3^w3 * b4^w4) ≤ w1 * b1 + w2 * b2 + w3 * b3 + w4 * b4 := by have g₀ : 0 < w1 := by rw [hw1] norm_num refine geom_mean_le_arith_mean4_weighted ?_ (by linarith) (by linarith) ?_ hb1 hb2 hb3 hb4 ?_ . exact le_of_lt g₀ . linarith . rw [hw4, hw3, hw2, hw1] norm_num repeat rw [mul_rpow _] . rw [hw4, hw3, hw2, hw1] at * refine le_trans h₀ ?_ ring_nf at * linarith repeat { assumption } . exact mul_nonneg hb1 hb2 . exact hb4 . refine mul_nonneg ?_ hb3 exact mul_nonneg hb1 hb2 lemma imo_2023_p4_2_2 (b1 b2 b3 b4 : ℝ) (hb1 : 0 ≤ b1) (hb2 : 0 ≤ b2) (hb3 : 0 ≤ b3) (hb4 : 0 ≤ b4) : -- (hw1 : w1 = (4:ℝ)⁻¹) -- (hw2 : w2 = w1) -- (hw3 : w3 = w2) -- (hw4 : w4 = w3) 0 ≤ b1 + b2 + b3 + b4 := by refine add_nonneg ?_ hb4 refine add_nonneg ?_ hb3 exact add_nonneg hb1 hb2 lemma imo_2023_p4_2_3 (b1 b2 b3 b4 : ℝ) (w1 w2 w3 w4 : ℝ) (hb1 : 0 ≤ b1) (hb2 : 0 ≤ b2) (hb3 : 0 ≤ b3) (hb4 : 0 ≤ b4) (hw1 : w1 = (4:ℝ)⁻¹) (hw2 : w2 = w1) (hw3 : w3 = w2) (hw4 : w4 = w3) : (b1 * b2 * b3 * b4) ^ ((4:ℝ)⁻¹) ≤ (b1 + b2 + b3 + b4) / 4 := by have h₀: (b1^w1 * b2^w2 * b3^w3 * b4^w4) ≤ w1 * b1 + w2 * b2 + w3 * b3 + w4 * b4 := by have g₀ : 0 < w1 := by rw [hw1] norm_num refine geom_mean_le_arith_mean4_weighted ?_ (by linarith) (by linarith) ?_ hb1 hb2 hb3 hb4 ?_ . exact le_of_lt g₀ . linarith . rw [hw4, hw3, hw2, hw1] norm_num repeat rw [mul_rpow _] . rw [hw4, hw3, hw2, hw1] at * refine le_trans h₀ ?_ ring_nf at * linarith repeat { assumption } . exact mul_nonneg hb1 hb2 . exact hb4 . refine mul_nonneg ?_ hb3 exact mul_nonneg hb1 hb2 lemma imo_2023_p4_2_4 (b1 b2 b3 b4 : ℝ) (w1 w2 w3 w4 : ℝ) (hb1 : 0 ≤ b1) (hb2 : 0 ≤ b2) (hb3 : 0 ≤ b3) (hb4 : 0 ≤ b4) (hw1 : w1 = (4:ℝ)⁻¹) (hw2 : w2 = w1) (hw3 : w3 = w2) (hw4 : w4 = w3) : b1 ^ w1 * b2 ^ w2 * b3 ^ w3 * b4 ^ w4 ≤ w1 * b1 + w2 * b2 + w3 * b3 + w4 * b4 := by have g₀ : 0 < w1 := by rw [hw1] norm_num refine geom_mean_le_arith_mean4_weighted ?_ (by linarith) (by linarith) ?_ hb1 hb2 hb3 hb4 ?_ . exact le_of_lt g₀ . linarith . rw [hw4, hw3, hw2, hw1] norm_num lemma imo_2023_p4_2_5 (b1 b2 b3 b4 : ℝ) (w1 w2 w3 w4 : ℝ) (hb1 : 0 ≤ b1) (hb2 : 0 ≤ b2) (hb3 : 0 ≤ b3) (hb4 : 0 ≤ b4) (hw1 : w1 = ((4:ℝ)⁻¹)) (hw2 : w2 = w1) (hw3 : w3 = w2) (hw4 : w4 = w3) (h₀ : b1 ^ w1 * b2 ^ w2 * b3 ^ w3 * b4 ^ w4 ≤ w1 * b1 + w2 * b2 + w3 * b3 + w4 * b4) : (b1 * b2 * b3 * b4) ^ (4:ℝ)⁻¹ ≤ (b1 + b2 + b3 + b4) / 4 := by repeat rw [mul_rpow _] . rw [hw4, hw3, hw2, hw1] at * refine le_trans h₀ ?_ ring_nf at * linarith repeat { assumption } . exact mul_nonneg hb1 hb2 . exact hb4 . refine mul_nonneg ?_ hb3 exact mul_nonneg hb1 hb2 lemma imo_2023_p4_2_6 (b1 b2 b3 b4 : ℝ) (w1 w2 w3 w4 : ℝ) -- (hb1 : 0 ≤ b1) -- (hb2 : 0 ≤ b2) -- (hb3 : 0 ≤ b3) -- (hb4 : 0 ≤ b4) (hw1 : w1 = ((4:ℝ)⁻¹)) (hw2 : w2 = w1) (hw3 : w3 = w2) (hw4 : w4 = w3) (h₀ : b1 ^ w1 * b2 ^ w2 * b3 ^ w3 * b4 ^ w4 ≤ w1 * b1 + w2 * b2 + w3 * b3 + w4 * b4) : b1 ^ (4:ℝ)⁻¹ * b2 ^ (4:ℝ)⁻¹ * b3 ^ (4:ℝ)⁻¹ * b4 ^ (4:ℝ)⁻¹ ≤ (b1 + b2 + b3 + b4) / 4 := by rw [hw4, hw3, hw2, hw1] at * refine le_trans h₀ ?_ ring_nf at * linarith lemma imo_2023_p4_3 (x a: ℕ → ℝ) (hxp: ∀ (i : ℕ), 0 < x i) (h₀: ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k))) (n: ℕ) (hn: 1 ≤ n ∧ n ≤ 2021) : (4 * a n ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) + (x (n + 1) + x (n + 2)) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by repeat rw [mul_add, add_mul] have g₁₁: 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1 := by refine le_of_lt ?_ refine Finset.sum_pos ?_ ?_ . exact fun i _ => hxp i . simp linarith have g₁₂: 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹ := by refine le_of_lt ?_ refine Finset.sum_pos ?_ ?_ . intros i _ exact inv_pos.mpr (hxp i) . simp linarith have h₃₂: 4 * ( ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) * (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) ) ^ (4:ℝ)⁻¹ ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by let b1:ℝ := (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) let b2:ℝ := (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) let b3:ℝ := (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) let b4:ℝ := x (n + 2) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) have hb1: b1 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) := by exact rfl have hb2: b2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) := by exact rfl have hb3: b3 = (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by exact rfl have hb4: b4 = x (n + 2) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by exact rfl rw [← hb1, ← hb2, ← hb3, ← hb4] have g₀: 4 * (b1 * b2 * b3 * b4) ^ (4:ℝ)⁻¹ ≤ b1 + b2 + b3 + b4 := by have b1p: 0 ≤ b1 := by rw [hb1] refine mul_nonneg ?_ ?_ . ring_nf exact g₁₁ . refine le_of_lt ?_ exact one_div_pos.mpr (hxp (n + 1)) have b2p: 0 ≤ b2 := by rw [hb2] refine mul_nonneg ?_ ?_ . ring_nf exact g₁₁ . refine le_of_lt ?_ exact one_div_pos.mpr (hxp (n + 2)) have b3p: 0 ≤ b3 := by rw [hb3] refine mul_nonneg ?_ ?_ . exact LT.lt.le (hxp (n + 1)) . ring_nf exact g₁₂ have b4p: 0 ≤ b4 := by rw [hb4] refine mul_nonneg ?_ ?_ . exact LT.lt.le (hxp (n + 2)) . ring_nf exact g₁₂ exact imo_2023_p4_2 b1 b2 b3 b4 b1p b2p b3p b4p linarith have h₃₃: 4 * a (n) = 4 * (((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) * (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) ) ^ (4:ℝ)⁻¹ := by simp ring_nf have g₀: (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 := by linarith have g₁: x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1 := by rw [mul_assoc] have gg₁: x (1 + n) * (x (1 + n))⁻¹ = 1 := by refine div_self ?_ exact ne_of_gt (hxp (1 + n)) have gg₂: x (2 + n) * (x (2 + n))⁻¹ = 1 := by refine div_self ?_ exact ne_of_gt (hxp (2 + n)) rw [gg₁, gg₂] norm_num rw [g₁] at g₀ rw [g₀] simp repeat rw [mul_rpow] . have g₂: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1/(2:ℝ)) := by rw [← rpow_mul g₁₁ (2:ℝ) (4:ℝ)⁻¹] norm_num have g₃: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1/(2:ℝ)) := by rw [← rpow_mul g₁₂ (2:ℝ) (4:ℝ)⁻¹] norm_num have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by refine h₀ n ?_ constructor . exact hn.1 . linarith norm_cast at * rw [g₂, g₃, ← mul_rpow g₁₁ g₁₂] rw [← sqrt_eq_rpow] ring_nf at g₄ exact g₄ . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) exact Eq.trans_le h₃₃ h₃₂ lemma imo_2023_p4_3_1 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) : 4 * a n ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by have g₁₁: 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1 := by refine le_of_lt ?_ refine Finset.sum_pos ?_ ?_ . exact fun i _ => hxp i . simp linarith have g₁₂: 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹ := by refine le_of_lt ?_ refine Finset.sum_pos ?_ ?_ . intros i _ exact inv_pos.mpr (hxp i) . simp linarith have h₃₂: 4 * ( ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) * (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) ) ^ (4:ℝ)⁻¹ ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by let b1:ℝ := (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) let b2:ℝ := (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) let b3:ℝ := (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) let b4:ℝ := x (n + 2) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) have hb1: b1 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) := by exact rfl have hb2: b2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) := by exact rfl have hb3: b3 = (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by exact rfl have hb4: b4 = x (n + 2) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by exact rfl rw [← hb1, ← hb2, ← hb3, ← hb4] have g₀: 4 * (b1 * b2 * b3 * b4) ^ (4:ℝ)⁻¹ ≤ b1 + b2 + b3 + b4 := by have b1p: 0 ≤ b1 := by rw [hb1] refine mul_nonneg ?_ ?_ . ring_nf exact g₁₁ . refine le_of_lt ?_ exact one_div_pos.mpr (hxp (n + 1)) have b2p: 0 ≤ b2 := by rw [hb2] refine mul_nonneg ?_ ?_ . ring_nf exact g₁₁ . refine le_of_lt ?_ exact one_div_pos.mpr (hxp (n + 2)) have b3p: 0 ≤ b3 := by rw [hb3] refine mul_nonneg ?_ ?_ . exact LT.lt.le (hxp (n + 1)) . ring_nf exact g₁₂ have b4p: 0 ≤ b4 := by rw [hb4] refine mul_nonneg ?_ ?_ . exact LT.lt.le (hxp (n + 2)) . ring_nf exact g₁₂ exact imo_2023_p4_2 b1 b2 b3 b4 b1p b2p b3p b4p linarith have h₃₃: 4 * a (n) = 4 * (((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) * (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) ) ^ (4:ℝ)⁻¹ := by simp ring_nf have g₀: (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 := by linarith have g₁: x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1 := by rw [mul_assoc] have gg₁: x (1 + n) * (x (1 + n))⁻¹ = 1 := by refine div_self ?_ exact ne_of_gt (hxp (1 + n)) have gg₂: x (2 + n) * (x (2 + n))⁻¹ = 1 := by refine div_self ?_ exact ne_of_gt (hxp (2 + n)) rw [gg₁, gg₂] norm_num rw [g₁] at g₀ rw [g₀] simp repeat rw [mul_rpow] . have g₂: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1/(2:ℝ)) := by rw [← rpow_mul g₁₁ (2:ℝ) (4:ℝ)⁻¹] norm_num have g₃: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1/(2:ℝ)) := by rw [← rpow_mul g₁₂ (2:ℝ) (4:ℝ)⁻¹] norm_num have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by refine h₀ n ?_ constructor . exact hn.1 . linarith norm_cast at * rw [g₂, g₃, ← mul_rpow g₁₁ g₁₂] rw [← sqrt_eq_rpow] ring_nf at g₄ exact g₄ . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) exact Eq.trans_le h₃₃ h₃₂ lemma imo_2023_p4_3_2 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1 := by refine le_of_lt ?_ refine Finset.sum_pos ?_ ?_ . exact fun i _ => hxp i . simp linarith lemma imo_2023_p4_3_3 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) : 4 * a n ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by have g₁₂: 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹ := by refine le_of_lt ?_ refine Finset.sum_pos ?_ ?_ . intros i _ exact inv_pos.mpr (hxp i) . simp linarith have h₃₂: 4 * ( ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) * (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) ) ^ (4:ℝ)⁻¹ ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by let b1:ℝ := (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) let b2:ℝ := (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) let b3:ℝ := (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) let b4:ℝ := x (n + 2) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) have hb1: b1 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) := by exact rfl have hb2: b2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) := by exact rfl have hb3: b3 = (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by exact rfl have hb4: b4 = x (n + 2) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by exact rfl rw [← hb1, ← hb2, ← hb3, ← hb4] have g₀: 4 * (b1 * b2 * b3 * b4) ^ (4:ℝ)⁻¹ ≤ b1 + b2 + b3 + b4 := by have b1p: 0 ≤ b1 := by rw [hb1] refine mul_nonneg ?_ ?_ . ring_nf exact g₁₁ . refine le_of_lt ?_ exact one_div_pos.mpr (hxp (n + 1)) have b2p: 0 ≤ b2 := by rw [hb2] refine mul_nonneg ?_ ?_ . ring_nf exact g₁₁ . refine le_of_lt ?_ exact one_div_pos.mpr (hxp (n + 2)) have b3p: 0 ≤ b3 := by rw [hb3] refine mul_nonneg ?_ ?_ . exact LT.lt.le (hxp (n + 1)) . ring_nf exact g₁₂ have b4p: 0 ≤ b4 := by rw [hb4] refine mul_nonneg ?_ ?_ . exact LT.lt.le (hxp (n + 2)) . ring_nf exact g₁₂ exact imo_2023_p4_2 b1 b2 b3 b4 b1p b2p b3p b4p linarith have h₃₃: 4 * a (n) = 4 * (((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) * (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) ) ^ (4:ℝ)⁻¹ := by simp ring_nf have g₀: (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 := by linarith have g₁: x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1 := by rw [mul_assoc] have gg₁: x (1 + n) * (x (1 + n))⁻¹ = 1 := by refine div_self ?_ exact ne_of_gt (hxp (1 + n)) have gg₂: x (2 + n) * (x (2 + n))⁻¹ = 1 := by refine div_self ?_ exact ne_of_gt (hxp (2 + n)) rw [gg₁, gg₂] norm_num rw [g₁] at g₀ rw [g₀] simp repeat rw [mul_rpow] . have g₂: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1/(2:ℝ)) := by rw [← rpow_mul g₁₁ (2:ℝ) (4:ℝ)⁻¹] norm_num have g₃: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1/(2:ℝ)) := by rw [← rpow_mul g₁₂ (2:ℝ) (4:ℝ)⁻¹] norm_num have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by refine h₀ n ?_ constructor . exact hn.1 . linarith norm_cast at * rw [g₂, g₃, ← mul_rpow g₁₁ g₁₂] rw [← sqrt_eq_rpow] ring_nf at g₄ exact g₄ . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) exact Eq.trans_le h₃₃ h₃₂ lemma imo_2023_p4_3_4 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) : -- (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹ := by refine le_of_lt ?_ refine Finset.sum_pos ?_ ?_ . intros i _ exact inv_pos.mpr (hxp i) . simp linarith lemma imo_2023_p4_3_5 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) : -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) : ∀ i ∈ Finset.Ico 1 (1 + n), 0 < (x i)⁻¹ := by intros i _ exact inv_pos.mpr (hxp i) lemma imo_2023_p4_3_6 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) : 4 * a n ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by have h₃₂: 4 * ( ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) * (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) ) ^ (4:ℝ)⁻¹ ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by let b1:ℝ := (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) let b2:ℝ := (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) let b3:ℝ := (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) let b4:ℝ := x (n + 2) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) have hb1: b1 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) := by exact rfl have hb2: b2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) := by exact rfl have hb3: b3 = (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by exact rfl have hb4: b4 = x (n + 2) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by exact rfl rw [← hb1, ← hb2, ← hb3, ← hb4] have g₀: 4 * (b1 * b2 * b3 * b4) ^ (4:ℝ)⁻¹ ≤ b1 + b2 + b3 + b4 := by have b1p: 0 ≤ b1 := by rw [hb1] refine mul_nonneg ?_ ?_ . ring_nf exact g₁₁ . refine le_of_lt ?_ exact one_div_pos.mpr (hxp (n + 1)) have b2p: 0 ≤ b2 := by rw [hb2] refine mul_nonneg ?_ ?_ . ring_nf exact g₁₁ . refine le_of_lt ?_ exact one_div_pos.mpr (hxp (n + 2)) have b3p: 0 ≤ b3 := by rw [hb3] refine mul_nonneg ?_ ?_ . exact LT.lt.le (hxp (n + 1)) . ring_nf exact g₁₂ have b4p: 0 ≤ b4 := by rw [hb4] refine mul_nonneg ?_ ?_ . exact LT.lt.le (hxp (n + 2)) . ring_nf exact g₁₂ exact imo_2023_p4_2 b1 b2 b3 b4 b1p b2p b3p b4p linarith have h₃₃: 4 * a (n) = 4 * (((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) * (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) ) ^ (4:ℝ)⁻¹ := by simp ring_nf have g₀: (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 := by linarith have g₁: x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1 := by rw [mul_assoc] have gg₁: x (1 + n) * (x (1 + n))⁻¹ = 1 := by refine div_self ?_ exact ne_of_gt (hxp (1 + n)) have gg₂: x (2 + n) * (x (2 + n))⁻¹ = 1 := by refine div_self ?_ exact ne_of_gt (hxp (2 + n)) rw [gg₁, gg₂] norm_num rw [g₁] at g₀ rw [g₀] simp repeat rw [mul_rpow] . have g₂: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1/(2:ℝ)) := by rw [← rpow_mul g₁₁ (2:ℝ) (4:ℝ)⁻¹] norm_num have g₃: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1/(2:ℝ)) := by rw [← rpow_mul g₁₂ (2:ℝ) (4:ℝ)⁻¹] norm_num have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by refine h₀ n ?_ constructor . exact hn.1 . linarith norm_cast at * rw [g₂, g₃, ← mul_rpow g₁₁ g₁₂] rw [← sqrt_eq_rpow] ring_nf at g₄ exact g₄ . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) exact Eq.trans_le h₃₃ h₃₂ lemma imo_2023_p4_3_7 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2021) (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) : 4 * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ (4:ℝ)⁻¹ ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by let b1:ℝ := (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) let b2:ℝ := (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) let b3:ℝ := (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) let b4:ℝ := x (n + 2) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) have hb1: b1 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) := by exact rfl have hb2: b2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) := by exact rfl have hb3: b3 = (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by exact rfl have hb4: b4 = x (n + 2) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by exact rfl rw [← hb1, ← hb2, ← hb3, ← hb4] have g₀: 4 * (b1 * b2 * b3 * b4) ^ (4:ℝ)⁻¹ ≤ b1 + b2 + b3 + b4 := by have b1p: 0 ≤ b1 := by rw [hb1] refine mul_nonneg ?_ ?_ . ring_nf exact g₁₁ . refine le_of_lt ?_ exact one_div_pos.mpr (hxp (n + 1)) have b2p: 0 ≤ b2 := by rw [hb2] refine mul_nonneg ?_ ?_ . ring_nf exact g₁₁ . refine le_of_lt ?_ exact one_div_pos.mpr (hxp (n + 2)) have b3p: 0 ≤ b3 := by rw [hb3] refine mul_nonneg ?_ ?_ . exact LT.lt.le (hxp (n + 1)) . ring_nf exact g₁₂ have b4p: 0 ≤ b4 := by rw [hb4] refine mul_nonneg ?_ ?_ . exact LT.lt.le (hxp (n + 2)) . ring_nf exact g₁₂ exact imo_2023_p4_2 b1 b2 b3 b4 b1p b2p b3p b4p linarith lemma imo_2023_p4_3_8 (x : ℕ → ℝ) (b1 b2 b3 b4 : ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2021) (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) (hb1 : b1 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) (hb2 : b2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) (hb3 : b3 = x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) (hb4 : b4 = x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) : 4 * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ (4:ℝ)⁻¹ ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by rw [← hb1, ← hb2, ← hb3, ← hb4] have g₀: 4 * (b1 * b2 * b3 * b4) ^ (4:ℝ)⁻¹ ≤ b1 + b2 + b3 + b4 := by have b1p: 0 ≤ b1 := by rw [hb1] refine mul_nonneg ?_ ?_ . ring_nf exact g₁₁ . refine le_of_lt ?_ exact one_div_pos.mpr (hxp (n + 1)) have b2p: 0 ≤ b2 := by rw [hb2] refine mul_nonneg ?_ ?_ . ring_nf exact g₁₁ . refine le_of_lt ?_ exact one_div_pos.mpr (hxp (n + 2)) have b3p: 0 ≤ b3 := by rw [hb3] refine mul_nonneg ?_ ?_ . exact LT.lt.le (hxp (n + 1)) . ring_nf exact g₁₂ have b4p: 0 ≤ b4 := by rw [hb4] refine mul_nonneg ?_ ?_ . exact LT.lt.le (hxp (n + 2)) . ring_nf exact g₁₂ exact imo_2023_p4_2 b1 b2 b3 b4 b1p b2p b3p b4p linarith lemma imo_2023_p4_3_9 (x : ℕ → ℝ) (b1 b2 b3 b4 : ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2021) (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) (hb1 : b1 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) (hb2 : b2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) (hb3 : b3 = x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) (hb4 : b4 = x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) : 4 * (b1 * b2 * b3 * b4) ^ (4:ℝ)⁻¹ ≤ b1 + b2 + (b3 + b4) := by have b1p: 0 ≤ b1 := by rw [hb1] refine mul_nonneg ?_ ?_ . ring_nf exact g₁₁ . refine le_of_lt ?_ exact one_div_pos.mpr (hxp (n + 1)) have b2p: 0 ≤ b2 := by rw [hb2] refine mul_nonneg ?_ ?_ . ring_nf exact g₁₁ . refine le_of_lt ?_ exact one_div_pos.mpr (hxp (n + 2)) have b3p: 0 ≤ b3 := by rw [hb3] refine mul_nonneg ?_ ?_ . exact LT.lt.le (hxp (n + 1)) . ring_nf exact g₁₂ have b4p: 0 ≤ b4 := by rw [hb4] refine mul_nonneg ?_ ?_ . exact LT.lt.le (hxp (n + 2)) . ring_nf exact g₁₂ rw [← add_assoc] exact imo_2023_p4_2 b1 b2 b3 b4 b1p b2p b3p b4p lemma imo_2023_p4_3_10 (x : ℕ → ℝ) (b1 : ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2021) (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) -- (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) (hb1 : b1 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) : -- (hb2 : b2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) -- (hb3 : b3 = x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (hb4 : b4 = x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) : 0 ≤ b1 := by rw [hb1] refine mul_nonneg ?_ ?_ . ring_nf exact g₁₁ . refine le_of_lt ?_ exact one_div_pos.mpr (hxp (n + 1)) lemma imo_2023_p4_3_11 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) : -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) -- (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) -- (hb1 : b1 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) -- (hb2 : b2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) -- (hb3 : b3 = x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (hb4 : b4 = x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) 0 ≤ 1 / x (n + 1) := by refine le_of_lt ?_ exact one_div_pos.mpr (hxp (n + 1)) lemma imo_2023_p4_3_12 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) (h₃₂ : 4 * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ (4:ℝ)⁻¹ ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) : 4 * a n ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by have h₃₃: 4 * a (n) = 4 * (((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) * (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) ) ^ (4:ℝ)⁻¹ := by simp ring_nf have g₀: (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 := by linarith have g₁: x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1 := by rw [mul_assoc] have gg₁: x (1 + n) * (x (1 + n))⁻¹ = 1 := by refine div_self ?_ exact ne_of_gt (hxp (1 + n)) have gg₂: x (2 + n) * (x (2 + n))⁻¹ = 1 := by refine div_self ?_ exact ne_of_gt (hxp (2 + n)) rw [gg₁, gg₂] norm_num rw [g₁] at g₀ rw [g₀] simp repeat rw [mul_rpow] . have g₂: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1/(2:ℝ)) := by rw [← rpow_mul g₁₁ (2:ℝ) (4:ℝ)⁻¹] norm_num have g₃: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1/(2:ℝ)) := by rw [← rpow_mul g₁₂ (2:ℝ) (4:ℝ)⁻¹] norm_num have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by refine h₀ n ?_ constructor . exact hn.1 . linarith norm_cast at * rw [g₂, g₃, ← mul_rpow g₁₁ g₁₂] rw [← sqrt_eq_rpow] ring_nf at g₄ exact g₄ . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) exact Eq.trans_le h₃₃ h₃₂ lemma imo_2023_p4_3_13 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) (h₃₂ : 4 * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ (4:ℝ)⁻¹ ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) : 4 * a n = 4 * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ (4:ℝ)⁻¹ := by simp ring_nf have g₀: (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 := by linarith have g₁: x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1 := by rw [mul_assoc] have gg₁: x (1 + n) * (x (1 + n))⁻¹ = 1 := by refine div_self ?_ exact ne_of_gt (hxp (1 + n)) have gg₂: x (2 + n) * (x (2 + n))⁻¹ = 1 := by refine div_self ?_ exact ne_of_gt (hxp (2 + n)) rw [gg₁, gg₂] norm_num rw [g₁] at g₀ rw [g₀] simp repeat rw [mul_rpow] . have g₂: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1/(2:ℝ)) := by rw [← rpow_mul g₁₁ (2:ℝ) (4:ℝ)⁻¹] norm_num have g₃: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1/(2:ℝ)) := by rw [← rpow_mul g₁₂ (2:ℝ) (4:ℝ)⁻¹] norm_num have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by refine h₀ n ?_ constructor . exact hn.1 . linarith norm_cast at * rw [g₂, g₃, ← mul_rpow g₁₁ g₁₂] rw [← sqrt_eq_rpow] ring_nf at g₄ exact g₄ . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) lemma imo_2023_p4_3_14 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) (h₃₂ : 4 * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ (4:ℝ)⁻¹ ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) : a n = ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) ^ (1 / (4:ℝ)) := by have g₀: (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 := by linarith have g₁: x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1 := by rw [mul_assoc] have gg₁: x (1 + n) * (x (1 + n))⁻¹ = 1 := by refine div_self ?_ exact ne_of_gt (hxp (1 + n)) have gg₂: x (2 + n) * (x (2 + n))⁻¹ = 1 := by refine div_self ?_ exact ne_of_gt (hxp (2 + n)) rw [gg₁, gg₂] norm_num rw [g₁] at g₀ rw [g₀] simp repeat rw [mul_rpow] have g₂: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1/(2:ℝ)) := by rw [← rpow_mul g₁₁ (2:ℝ) (4:ℝ)⁻¹] norm_num have g₃: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1/(2:ℝ)) := by rw [← rpow_mul g₁₂ (2:ℝ) (4:ℝ)⁻¹] norm_num have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by refine h₀ n ?_ constructor . exact hn.1 . linarith norm_cast at * rw [g₂, g₃] rw [← mul_rpow] rw [← sqrt_eq_rpow] ring_nf at g₄ exact g₄ . exact g₁₁ . exact g₁₂ . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) lemma imo_2023_p4_3_15 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) (h₃₂ : 4 * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ (4:ℝ)⁻¹ ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) : a n = ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) ^ (1 / (4:ℝ)) := by have g₁: x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1 := by rw [mul_assoc] have gg₁: x (1 + n) * (x (1 + n))⁻¹ = 1 := by refine div_self ?_ exact ne_of_gt (hxp (1 + n)) have gg₂: x (2 + n) * (x (2 + n))⁻¹ = 1 := by refine div_self ?_ exact ne_of_gt (hxp (2 + n)) rw [gg₁, gg₂] norm_num rw [g₁] at g₀ rw [g₀] simp repeat rw [mul_rpow] have g₂: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1/(2:ℝ)) := by rw [← rpow_mul g₁₁ (2:ℝ) (4:ℝ)⁻¹] norm_num have g₃: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1/(2:ℝ)) := by rw [← rpow_mul g₁₂ (2:ℝ) (4:ℝ)⁻¹] norm_num have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by refine h₀ n ?_ constructor . exact hn.1 . linarith norm_cast at * rw [g₂, g₃] rw [← mul_rpow] rw [← sqrt_eq_rpow] ring_nf at g₄ exact g₄ . exact g₁₁ . exact g₁₂ . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) lemma imo_2023_p4_3_16 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) : -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) -- (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) -- (h₃₂ : 4 * -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * -- (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * -- (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ -- (4:ℝ)⁻¹ ≤ -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + -- ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = -- x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) : x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1 := by rw [mul_assoc] have gg₁: x (1 + n) * (x (1 + n))⁻¹ = 1 := by refine div_self ?_ exact ne_of_gt (hxp (1 + n)) have gg₂: x (2 + n) * (x (2 + n))⁻¹ = 1 := by refine div_self ?_ exact ne_of_gt (hxp (2 + n)) rw [gg₁, gg₂] norm_num lemma imo_2023_p4_3_17 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) : -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) -- (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) -- (h₃₂ : 4 * -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * -- (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * -- (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ -- (4:ℝ)⁻¹ ≤ -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + -- ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = -- x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) : x (1 + n) * (x (1 + n))⁻¹ = 1 := by refine div_self ?_ exact ne_of_gt (hxp (1 + n)) lemma imo_2023_p4_3_18 (x : ℕ → ℝ) -- (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) -- (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) -- (h₃₂ : 4 * -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * -- (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * -- (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ -- (4:ℝ)⁻¹ ≤ -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + -- ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = -- x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) (gg₁ : x (1 + n) * (x (1 + n))⁻¹ = 1) (gg₂ : x (2 + n) * (x (2 + n))⁻¹ = 1) : x (1 + n) * (x (1 + n))⁻¹ * (x (2 + n) * (x (2 + n))⁻¹) = 1 := by rw [gg₁, gg₂] norm_num lemma imo_2023_p4_3_19 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) (h₃₂ : 4 * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ (4:ℝ)⁻¹ ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) (g₁ : x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1) : a n = ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) ^ (1 / (4:ℝ)) := by rw [g₁] at g₀ rw [g₀] simp repeat rw [mul_rpow] have g₂: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1/(2:ℝ)) := by rw [← rpow_mul g₁₁ (2:ℝ) (4:ℝ)⁻¹] norm_num have g₃: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1/(2:ℝ)) := by rw [← rpow_mul g₁₂ (2:ℝ) (4:ℝ)⁻¹] norm_num have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by refine h₀ n ?_ constructor . exact hn.1 . linarith norm_cast at * rw [g₂, g₃, ← mul_rpow] rw [← sqrt_eq_rpow] ring_nf at g₄ exact g₄ . exact g₁₁ . exact g₁₂ . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) lemma imo_2023_p4_3_20 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) (h₃₂ : 4 * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ (4:ℝ)⁻¹ ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * (x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹) * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = 1 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) (g₁ : x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1) : a n = ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) ^ (4:ℝ)⁻¹ := by repeat rw [mul_rpow] have g₂: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1/(2:ℝ)) := by rw [← rpow_mul g₁₁ (2:ℝ) (4:ℝ)⁻¹] norm_num have g₃: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1/(2:ℝ)) := by rw [← rpow_mul g₁₂ (2:ℝ) (4:ℝ)⁻¹] norm_num have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by refine h₀ n ?_ constructor . exact hn.1 . linarith norm_cast at * rw [g₂, g₃, ← mul_rpow] rw [← sqrt_eq_rpow] ring_nf at g₄ exact g₄ . exact g₁₁ . exact g₁₂ . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) lemma imo_2023_p4_3_21 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) (h₃₂ : 4 * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ (4:ℝ)⁻¹ ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = 1 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) (g₁ : x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1) (g₂ : ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1/(2:ℝ))) (g₃ : ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1/(2:ℝ))) : a n = ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2) ^ (4:ℝ)⁻¹ * ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) ^ (4:ℝ)⁻¹ := by have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by refine h₀ n ?_ constructor . exact hn.1 . linarith norm_cast at * rw [g₂, g₃] rw [← mul_rpow] rw [← sqrt_eq_rpow] ring_nf at g₄ exact g₄ . exact g₁₁ . exact g₁₂ lemma imo_2023_p4_3_22 (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) : -- (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) -- (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) -- (h₃₂ : 4 * -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * -- (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * -- (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ -- (4:ℝ)⁻¹ ≤ -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + -- ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = -- 1 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) -- (g₁ : x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1) -- (g₂ : ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2) ^ (4:ℝ)⁻¹ = -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1 / 2)) -- (g₃ : ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) ^ (4:ℝ)⁻¹ = -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1 / 2)) : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by refine h₀ n ?_ constructor . exact hn.1 . linarith norm_cast lemma imo_2023_p4_3_23 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2021) (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) (h₃₂ : 4 * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ (4:ℝ)⁻¹ ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = 1 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) (g₁ : x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1) (g₂ : ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1 / (2:ℝ))) (g₃ : ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1 / (2:ℝ))) (g₄ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) : a n = ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2) ^ (4:ℝ)⁻¹ * ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) ^ (4:ℝ)⁻¹ := by norm_cast at * rw [g₂, g₃, ← mul_rpow] . rw [← sqrt_eq_rpow] ring_nf at g₄ exact g₄ . exact g₁₁ . exact g₁₂ lemma imo_2023_p4_3_24 (x : ℕ → ℝ) (a : ℕ → ℝ) (n : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1)) (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) -- (h₃₂ : 4 * -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * -- (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1) * -- (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1)) ^ -- (4:ℝ)⁻¹ ≤ -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + -- ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1) + -- x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1)) -- (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = -- 1 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) -- (g₁ : x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1) (g₂ : ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1 / (2:ℝ))) (g₃ : ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) ^ (4:ℝ)⁻¹ = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1 / (2:ℝ))) (g₄ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1)) : a n = ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2) ^ (4:ℝ)⁻¹ * ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) ^ (4:ℝ)⁻¹ := by rw [g₂, g₃, ← mul_rpow] . rw [← sqrt_eq_rpow] ring_nf at g₄ exact g₄ . exact g₁₁ . exact g₁₂ lemma imo_2023_p4_3_25 (x : ℕ → ℝ) (a : ℕ → ℝ) (n : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1) ) -- (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) -- (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) -- (h₃₂ : 4 * -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * -- (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1) * -- (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1)) ^ -- (4:ℝ)⁻¹ ≤ -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + -- ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1) + -- x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1)) -- (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = -- 1 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) -- (g₁ : x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1) -- (g₂ : ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2) ^ (4:ℝ)⁻¹ = -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1 / 2)) -- (g₃ : ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) ^ (4:ℝ)⁻¹ = -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1 / 2)) (g₄ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1)) : a n = ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) * Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1 / (2:ℝ)) := by rw [← sqrt_eq_rpow] ring_nf at g₄ exact g₄ lemma imo_2023_p4_3_26 (x : ℕ → ℝ) (a : ℕ → ℝ) (n : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1)) -- (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) -- (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) -- (h₃₂ : 4 * -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * -- (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1) * -- (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1)) ^ -- (4:ℝ)⁻¹ ≤ -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + -- ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1) + -- x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1)) -- (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = -- 1 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) -- (g₁ : x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1) -- (g₂ : ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2) ^ (4:ℝ)⁻¹ = -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1 / 2)) -- (g₃ : ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) ^ (4:ℝ)⁻¹ = -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1 / 2)) (g₄ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1)) : a n = √((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) * Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) := by ring_nf at g₄ exact g₄ lemma imo_2023_p4_3_27 (x : ℕ → ℝ) -- (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) : -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) -- (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) -- (h₃₂ : 4 * -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * -- (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * -- (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ -- (4:ℝ)⁻¹ ≤ -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + -- ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = -- 1 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) -- (g₁ : x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1) : 0 ≤ (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 := by exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) lemma imo_2023_p4_3_28 (x : ℕ → ℝ) -- (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) : -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) -- (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) -- (h₃₂ : 4 * -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * -- (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * -- (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ -- (4:ℝ)⁻¹ ≤ -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + -- ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = -- 1 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) -- (g₁ : x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1) : 0 ≤ (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 := by exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) lemma imo_2023_p4_3_29 (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (n : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) -- (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) (h₃₂ : 4 * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ (4:ℝ)⁻¹ ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (h₃₃ : 4 * a n = 4 * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ (4:ℝ)⁻¹) : 4 * a n ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by exact Eq.trans_le h₃₃ h₃₂ lemma imo_2023_p4_4 (x a: ℕ → ℝ) (hxp: ∀ (i : ℕ), 0 < x i) (hx: ∀ (i j : ℕ), i ≠ j → x i ≠ x j) (h₀: ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k))) (h₀₁: ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) : (∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) := by intros n hn have g₀: 0 ≤ a n + 2 := by refine le_of_lt ?_ refine add_pos ?_ (by norm_num) refine h₀₁ n ?_ constructor . exact hn.1 . linarith have g₁: 0 ≤ a (n + 2) := by refine le_of_lt ?_ refine h₀₁ (n + 2) ?_ constructor . linarith . linarith rw [← sqrt_sq g₀, ← sqrt_sq g₁] have g₂: 0 ≤ (a n + 2) ^ 2 := by exact sq_nonneg (a n + 2) simp refine Real.sqrt_lt_sqrt g₂ ?_ have g₃: 0 ≤ ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) := by refine le_of_lt ?_ refine mul_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . exact fun i _ => hxp i . simp linarith . refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith have gn₀: a (n) ^ 2 = ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) := by rw [← sq_sqrt g₃] have g₄: 0 ≤ a n := by refine le_of_lt ?_ refine h₀₁ n ?_ constructor . exact hn.1 . linarith refine (sq_eq_sq₀ g₄ ?_).mpr ?_ . exact sqrt_nonneg ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) refine h₀ (n) ?_ constructor . exact hn.1 . linarith have gn₁: a (n + 2) = sqrt ((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k) := by refine h₀ (n + 2) ?_ constructor . linarith . linarith rw [add_sq, gn₁, sq_sqrt] . have ga₀: 1 ≤ n + 2 := by linarith rw [Finset.sum_Ico_succ_top ga₀ _, Finset.sum_Ico_succ_top ga₀ _] have ga₁: 1 ≤ n + 1 := by linarith rw [Finset.sum_Ico_succ_top ga₁ _, Finset.sum_Ico_succ_top ga₁ _] rw [add_assoc, add_assoc, add_assoc] rw [add_mul, mul_add] rw [← gn₀] repeat rw [add_assoc] refine add_lt_add_left ?_ (a (n) ^ 2) rw [mul_add (x (n + 1) + x (n + 2))] have h₂: 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)) := by repeat rw [add_mul, mul_add, mul_add] repeat rw [mul_div_left_comm _ 1 _, one_mul] repeat rw [div_self ?_] . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1) := by rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 2)) have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2) := by rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 1)) have h₂₀: 0 < x (n + 1) * x (n + 2) := by refine mul_pos ?_ ?_ . exact hxp (n + 1) . exact hxp (n + 2) have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) rw [mul_add, hc₂, hc₃, ← sq, ← sq] refine lt_of_sub_pos ?_ have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2 := by rw [sub_sq] linarith rw [gh₂₁] refine (sq_pos_iff).mpr ?_ refine sub_ne_zero.mpr ?_ exact hx (n+1) (n+2) (by linarith) linarith . exact ne_of_gt (hxp (n + 2)) . exact ne_of_gt (hxp (n + 1)) clear gn₀ gn₁ g₀ g₁ g₂ g₃ ga₀ ga₁ have h₃: 4 * a (n) ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) + ((x (n + 1) + x (n + 2)) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by exact imo_2023_p4_3 (fun k => x k) a hxp h₀ n hn linarith . refine mul_nonneg ?_ ?_ . refine Finset.sum_nonneg ?_ intros i _ exact LT.lt.le (hxp i) . refine Finset.sum_nonneg ?_ intros i _ simp exact LT.lt.le (hxp i) lemma imo_2023_p4_4_1 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) : 0 ≤ a n + 2 := by refine le_of_lt ?_ refine add_pos ?_ (by norm_num) refine h₀₁ n ?_ constructor . exact hn.1 . linarith lemma imo_2023_p4_4_2 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) : 0 < a n := by refine h₀₁ n ?_ constructor . exact hn.1 . linarith lemma imo_2023_p4_4_3 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) : -- (g₀ : 0 ≤ a n + 2) : 0 ≤ a (n + 2) := by refine le_of_lt ?_ refine h₀₁ (n + 2) ?_ constructor . linarith . linarith lemma imo_2023_p4_4_4 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) (g₀ : 0 ≤ a n + 2) (g₁ : 0 ≤ a (n + 2)) : a n + 2 < a (n + 2) := by rw [← sqrt_sq g₀, ← sqrt_sq g₁] have g₂: 0 ≤ (a n + 2) ^ 2 := by exact sq_nonneg (a n + 2) simp refine Real.sqrt_lt_sqrt g₂ ?_ have g₃: 0 ≤ ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) := by refine le_of_lt ?_ refine mul_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . exact fun i _ => hxp i . simp linarith . refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith have gn₀: a (n) ^ 2 = ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) := by rw [← sq_sqrt g₃] have g₄: 0 ≤ a n := by refine le_of_lt ?_ refine h₀₁ n ?_ constructor . exact hn.1 . linarith refine (sq_eq_sq₀ g₄ ?_).mpr ?_ . exact sqrt_nonneg ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) refine h₀ (n) ?_ constructor . exact hn.1 . linarith have gn₁: a (n + 2) = sqrt ((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k) := by refine h₀ (n + 2) ?_ constructor . linarith . linarith rw [add_sq, gn₁, sq_sqrt] . have ga₀: 1 ≤ n + 2 := by linarith rw [Finset.sum_Ico_succ_top ga₀ _, Finset.sum_Ico_succ_top ga₀ _] have ga₁: 1 ≤ n + 1 := by linarith rw [Finset.sum_Ico_succ_top ga₁ _, Finset.sum_Ico_succ_top ga₁ _] rw [add_assoc, add_assoc, add_assoc] rw [add_mul, mul_add] rw [← gn₀] repeat rw [add_assoc] refine add_lt_add_left ?_ (a (n) ^ 2) rw [mul_add (x (n + 1) + x (n + 2))] have h₂: 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)) := by repeat rw [add_mul, mul_add, mul_add] repeat rw [mul_div_left_comm _ 1 _, one_mul] repeat rw [div_self ?_] . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1) := by rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 2)) have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2) := by rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 1)) have h₂₀: 0 < x (n + 1) * x (n + 2) := by refine mul_pos ?_ ?_ . exact hxp (n + 1) . exact hxp (n + 2) have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) rw [mul_add, hc₂, hc₃, ← sq, ← sq] refine lt_of_sub_pos ?_ have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2 := by rw [sub_sq] linarith rw [gh₂₁] refine (sq_pos_iff).mpr ?_ refine sub_ne_zero.mpr ?_ exact hx (n+1) (n+2) (by linarith) linarith . exact ne_of_gt (hxp (n + 2)) . exact ne_of_gt (hxp (n + 1)) clear gn₀ gn₁ g₀ g₁ g₂ g₃ ga₀ ga₁ have h₃: 4 * a (n) ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) + ((x (n + 1) + x (n + 2)) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by exact imo_2023_p4_3 (fun k => x k) a hxp h₀ n hn linarith . refine mul_nonneg ?_ ?_ . refine Finset.sum_nonneg ?_ intros i _ exact LT.lt.le (hxp i) . refine Finset.sum_nonneg ?_ intros i _ simp exact LT.lt.le (hxp i) lemma imo_2023_p4_4_5 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) (g₀ : 0 ≤ a n + 2) (g₁ : 0 ≤ a (n + 2)) : √((a n + 2) ^ 2) < √(a (n + 2) ^ 2) := by have g₂: 0 ≤ (a n + 2) ^ 2 := by exact sq_nonneg (a n + 2) simp refine Real.sqrt_lt_sqrt g₂ ?_ have g₃: 0 ≤ ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) := by refine le_of_lt ?_ refine mul_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . exact fun i _ => hxp i . simp linarith . refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith have gn₀: a (n) ^ 2 = ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) := by rw [← sq_sqrt g₃] have g₄: 0 ≤ a n := by refine le_of_lt ?_ refine h₀₁ n ?_ constructor . exact hn.1 . linarith refine (sq_eq_sq₀ g₄ ?_).mpr ?_ . exact sqrt_nonneg ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) refine h₀ (n) ?_ constructor . exact hn.1 . linarith have gn₁: a (n + 2) = sqrt ((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k) := by refine h₀ (n + 2) ?_ constructor . linarith . linarith rw [add_sq, gn₁, sq_sqrt] . have ga₀: 1 ≤ n + 2 := by linarith rw [Finset.sum_Ico_succ_top ga₀ _, Finset.sum_Ico_succ_top ga₀ _] have ga₁: 1 ≤ n + 1 := by linarith rw [Finset.sum_Ico_succ_top ga₁ _, Finset.sum_Ico_succ_top ga₁ _] rw [add_assoc, add_assoc, add_assoc] rw [add_mul, mul_add] rw [← gn₀] repeat rw [add_assoc] refine add_lt_add_left ?_ (a (n) ^ 2) rw [mul_add (x (n + 1) + x (n + 2))] have h₂: 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)) := by repeat rw [add_mul, mul_add, mul_add] repeat rw [mul_div_left_comm _ 1 _, one_mul] repeat rw [div_self ?_] . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1) := by rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 2)) have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2) := by rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 1)) have h₂₀: 0 < x (n + 1) * x (n + 2) := by refine mul_pos ?_ ?_ . exact hxp (n + 1) . exact hxp (n + 2) have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) rw [mul_add, hc₂, hc₃, ← sq, ← sq] refine lt_of_sub_pos ?_ have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2 := by rw [sub_sq] linarith rw [gh₂₁] refine (sq_pos_iff).mpr ?_ refine sub_ne_zero.mpr ?_ exact hx (n+1) (n+2) (by linarith) linarith . exact ne_of_gt (hxp (n + 2)) . exact ne_of_gt (hxp (n + 1)) clear gn₀ gn₁ g₀ g₁ g₂ g₃ ga₀ ga₁ have h₃: 4 * a (n) ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) + ((x (n + 1) + x (n + 2)) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by exact imo_2023_p4_3 (fun k => x k) a hxp h₀ n hn linarith . refine mul_nonneg ?_ ?_ . refine Finset.sum_nonneg ?_ intros i _ exact LT.lt.le (hxp i) . refine Finset.sum_nonneg ?_ intros i _ simp exact LT.lt.le (hxp i) lemma imo_2023_p4_4_6 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) (g₀ : 0 ≤ a n + 2) (g₁ : 0 ≤ a (n + 2)) (g₂ : 0 ≤ (a n + 2) ^ 2) : (a n + 2) ^ 2 < a (n + 2) ^ 2 := by have g₃: 0 ≤ ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) := by refine le_of_lt ?_ refine mul_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . exact fun i _ => hxp i . simp linarith . refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith have gn₀: a (n) ^ 2 = ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) := by rw [← sq_sqrt g₃] have g₄: 0 ≤ a n := by refine le_of_lt ?_ refine h₀₁ n ?_ constructor . exact hn.1 . linarith refine (sq_eq_sq₀ g₄ ?_).mpr ?_ . exact sqrt_nonneg ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) . refine h₀ (n) ?_ constructor . exact hn.1 . linarith have gn₁: a (n + 2) = sqrt ((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k) := by refine h₀ (n + 2) ?_ constructor . linarith . linarith rw [add_sq, gn₁, sq_sqrt] . have ga₀: 1 ≤ n + 2 := by linarith rw [Finset.sum_Ico_succ_top ga₀ _, Finset.sum_Ico_succ_top ga₀ _] have ga₁: 1 ≤ n + 1 := by linarith rw [Finset.sum_Ico_succ_top ga₁ _, Finset.sum_Ico_succ_top ga₁ _] rw [add_assoc, add_assoc, add_assoc] rw [add_mul, mul_add] rw [← gn₀] repeat rw [add_assoc] refine add_lt_add_left ?_ (a (n) ^ 2) rw [mul_add (x (n + 1) + x (n + 2))] have h₂: 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)) := by repeat rw [add_mul, mul_add, mul_add] repeat rw [mul_div_left_comm _ 1 _, one_mul] repeat rw [div_self ?_] . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1) := by rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 2)) have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2) := by rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 1)) have h₂₀: 0 < x (n + 1) * x (n + 2) := by refine mul_pos ?_ ?_ . exact hxp (n + 1) . exact hxp (n + 2) have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) rw [mul_add, hc₂, hc₃, ← sq, ← sq] refine lt_of_sub_pos ?_ have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2 := by rw [sub_sq] linarith rw [gh₂₁] refine (sq_pos_iff).mpr ?_ refine sub_ne_zero.mpr ?_ exact hx (n+1) (n+2) (by linarith) linarith . exact ne_of_gt (hxp (n + 2)) . exact ne_of_gt (hxp (n + 1)) clear gn₀ gn₁ g₀ g₁ g₂ g₃ ga₀ ga₁ have h₃: 4 * a (n) ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) + ((x (n + 1) + x (n + 2)) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by exact imo_2023_p4_3 (fun k => x k) a hxp h₀ n hn linarith . refine mul_nonneg ?_ ?_ . refine Finset.sum_nonneg ?_ intros i _ exact LT.lt.le (hxp i) . refine Finset.sum_nonneg ?_ intros i _ simp exact LT.lt.le (hxp i) lemma imo_2023_p4_4_7 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) : -- (g₀ : 0 ≤ a n + 2) -- (g₁ : 0 ≤ a (n + 2)) -- (g₂ : 0 ≤ (a n + 2) ^ 2) : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k := by refine le_of_lt ?_ refine mul_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . exact fun i _ => hxp i . simp linarith . refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith lemma imo_2023_p4_4_8 (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₀ : 0 ≤ a n + 2) -- (g₁ : 0 ≤ a (n + 2)) -- (g₂ : 0 ≤ (a n + 2) ^ 2) (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k := by rw [← sq_sqrt g₃] have g₄: 0 ≤ a n := by refine le_of_lt ?_ refine h₀₁ n ?_ constructor . exact hn.1 . linarith refine (sq_eq_sq₀ g₄ ?_).mpr ?_ . exact sqrt_nonneg ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) refine h₀ (n) ?_ constructor . exact hn.1 . linarith lemma imo_2023_p4_4_9 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) (g₀ : 0 ≤ a n + 2) (g₁ : 0 ≤ a (n + 2)) (g₂ : 0 ≤ (a n + 2) ^ 2) (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) : (a n + 2) ^ 2 < a (n + 2) ^ 2 := by have gn₀: a (n) ^ 2 = ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) := by rw [← sq_sqrt g₃] have g₄: 0 ≤ a n := by refine le_of_lt ?_ refine h₀₁ n ?_ constructor . exact hn.1 . linarith refine (sq_eq_sq₀ g₄ ?_).mpr ?_ . exact sqrt_nonneg ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) . refine h₀ (n) ?_ constructor . exact hn.1 . linarith have gn₁: a (n + 2) = sqrt ((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k) := by refine h₀ (n + 2) ?_ constructor . linarith . linarith rw [add_sq, gn₁, sq_sqrt] . have ga₀: 1 ≤ n + 2 := by linarith rw [Finset.sum_Ico_succ_top ga₀ _, Finset.sum_Ico_succ_top ga₀ _] have ga₁: 1 ≤ n + 1 := by linarith rw [Finset.sum_Ico_succ_top ga₁ _, Finset.sum_Ico_succ_top ga₁ _] rw [add_assoc, add_assoc, add_assoc] rw [add_mul, mul_add] rw [← gn₀] repeat rw [add_assoc] refine add_lt_add_left ?_ (a (n) ^ 2) rw [mul_add (x (n + 1) + x (n + 2))] have h₂: 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)) := by repeat rw [add_mul, mul_add, mul_add] repeat rw [mul_div_left_comm _ 1 _, one_mul] repeat rw [div_self ?_] . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1) := by rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 2)) have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2) := by rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 1)) have h₂₀: 0 < x (n + 1) * x (n + 2) := by refine mul_pos ?_ ?_ . exact hxp (n + 1) . exact hxp (n + 2) have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) rw [mul_add, hc₂, hc₃, ← sq, ← sq] refine lt_of_sub_pos ?_ have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2 := by rw [sub_sq] linarith rw [gh₂₁] refine (sq_pos_iff).mpr ?_ refine sub_ne_zero.mpr ?_ exact hx (n+1) (n+2) (by linarith) linarith . exact ne_of_gt (hxp (n + 2)) . exact ne_of_gt (hxp (n + 1)) clear gn₀ gn₁ g₀ g₁ g₂ g₃ ga₀ ga₁ have h₃: 4 * a (n) ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) + ((x (n + 1) + x (n + 2)) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by exact imo_2023_p4_3 (fun k => x k) a hxp h₀ n hn linarith . refine mul_nonneg ?_ ?_ . refine Finset.sum_nonneg ?_ intros i _ exact LT.lt.le (hxp i) . refine Finset.sum_nonneg ?_ intros i _ simp exact LT.lt.le (hxp i) lemma imo_2023_p4_4_10 (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₀ : 0 ≤ a n + 2) -- (g₁ : 0 ≤ a (n + 2)) -- (g₂ : 0 ≤ (a n + 2) ^ 2) -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) (g₄ : 0 ≤ a n) : a n ^ 2 = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) ^ 2 := by refine (sq_eq_sq₀ g₄ ?_).mpr ?_ . exact sqrt_nonneg ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) . refine h₀ (n) ?_ constructor . exact hn.1 . linarith lemma imo_2023_p4_4_11 (x : ℕ → ℝ) -- (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) : -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₀ : 0 ≤ a n + 2) -- (g₁ : 0 ≤ a (n + 2)) -- (g₂ : 0 ≤ (a n + 2) ^ 2) -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (g₄ : 0 ≤ a n) : 0 ≤ √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by exact sqrt_nonneg ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) lemma imo_2023_p4_4_12 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) (g₀ : 0 ≤ a n + 2) (g₁ : 0 ≤ a (n + 2)) (g₂ : 0 ≤ (a n + 2) ^ 2) (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) (gn₁ : a (n + 2) = √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) : (a n + 2) ^ 2 < a (n + 2) ^ 2 := by rw [add_sq, gn₁, sq_sqrt] . have ga₀: 1 ≤ n + 2 := by linarith rw [Finset.sum_Ico_succ_top ga₀ _, Finset.sum_Ico_succ_top ga₀ _] have ga₁: 1 ≤ n + 1 := by linarith rw [Finset.sum_Ico_succ_top ga₁ _, Finset.sum_Ico_succ_top ga₁ _] rw [add_assoc, add_assoc, add_assoc] rw [add_mul, mul_add] rw [← gn₀] repeat rw [add_assoc] refine add_lt_add_left ?_ (a (n) ^ 2) rw [mul_add (x (n + 1) + x (n + 2))] have h₂: 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)) := by repeat rw [add_mul, mul_add, mul_add] repeat rw [mul_div_left_comm _ 1 _, one_mul] repeat rw [div_self ?_] . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1) := by rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 2)) have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2) := by rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 1)) have h₂₀: 0 < x (n + 1) * x (n + 2) := by refine mul_pos ?_ ?_ . exact hxp (n + 1) . exact hxp (n + 2) have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) rw [mul_add, hc₂, hc₃, ← sq, ← sq] refine lt_of_sub_pos ?_ have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2 := by rw [sub_sq] linarith rw [gh₂₁] refine (sq_pos_iff).mpr ?_ refine sub_ne_zero.mpr ?_ exact hx (n+1) (n+2) (by linarith) linarith . exact ne_of_gt (hxp (n + 2)) . exact ne_of_gt (hxp (n + 1)) clear gn₀ gn₁ g₀ g₁ g₂ g₃ ga₀ ga₁ have h₃: 4 * a (n) ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) + ((x (n + 1) + x (n + 2)) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by exact imo_2023_p4_3 (fun k => x k) a hxp h₀ n hn linarith . refine mul_nonneg ?_ ?_ . refine Finset.sum_nonneg ?_ intros i _ exact LT.lt.le (hxp i) . refine Finset.sum_nonneg ?_ intros i _ simp exact LT.lt.le (hxp i) lemma imo_2023_p4_4_13 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) (g₀ : 0 ≤ a n + 2) (g₁ : 0 ≤ a (n + 2)) (g₂ : 0 ≤ (a n + 2) ^ 2) (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) (gn₁ : a (n + 2) = √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) : a n ^ 2 + 2 * a n * 2 + 2 ^ 2 < (Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k := by have ga₀: 1 ≤ n + 2 := by linarith rw [Finset.sum_Ico_succ_top ga₀ _, Finset.sum_Ico_succ_top ga₀ _] have ga₁: 1 ≤ n + 1 := by linarith rw [Finset.sum_Ico_succ_top ga₁ _, Finset.sum_Ico_succ_top ga₁ _] rw [add_assoc, add_assoc, add_assoc] rw [add_mul, mul_add] rw [← gn₀] repeat rw [add_assoc] refine add_lt_add_left ?_ (a (n) ^ 2) rw [mul_add (x (n + 1) + x (n + 2))] have h₂: 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)) := by repeat rw [add_mul, mul_add, mul_add] repeat rw [mul_div_left_comm _ 1 _, one_mul] repeat rw [div_self ?_] . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1) := by rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 2)) have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2) := by rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 1)) have h₂₀: 0 < x (n + 1) * x (n + 2) := by refine mul_pos ?_ ?_ . exact hxp (n + 1) . exact hxp (n + 2) have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) rw [mul_add, hc₂, hc₃, ← sq, ← sq] refine lt_of_sub_pos ?_ have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2 := by rw [sub_sq] linarith rw [gh₂₁] refine (sq_pos_iff).mpr ?_ refine sub_ne_zero.mpr ?_ exact hx (n+1) (n+2) (by linarith) linarith . exact ne_of_gt (hxp (n + 2)) . exact ne_of_gt (hxp (n + 1)) clear gn₀ gn₁ g₀ g₁ g₂ g₃ ga₀ ga₁ have h₃: 4 * a (n) ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) + ((x (n + 1) + x (n + 2)) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by exact imo_2023_p4_3 (fun k => x k) a hxp h₀ n hn linarith lemma imo_2023_p4_4_14 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) (g₀ : 0 ≤ a n + 2) (g₁ : 0 ≤ a (n + 2)) (g₂ : 0 ≤ (a n + 2) ^ 2) (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) (gn₁ : a (n + 2) = √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) (ga₀ : 1 ≤ n + 2) : a n ^ 2 + 2 * a n * 2 + 2 ^ 2 < ((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) + x (n + 2)) * ((Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k) + 1 / x (n + 2)) := by have ga₁: 1 ≤ n + 1 := by linarith rw [Finset.sum_Ico_succ_top ga₁ _, Finset.sum_Ico_succ_top ga₁ _] rw [add_assoc, add_assoc, add_assoc] rw [add_mul, mul_add] rw [← gn₀] repeat rw [add_assoc] refine add_lt_add_left ?_ (a (n) ^ 2) rw [mul_add (x (n + 1) + x (n + 2))] have h₂: 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)) := by repeat rw [add_mul, mul_add, mul_add] repeat rw [mul_div_left_comm _ 1 _, one_mul] repeat rw [div_self ?_] . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1) := by rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 2)) have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2) := by rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 1)) have h₂₀: 0 < x (n + 1) * x (n + 2) := by refine mul_pos ?_ ?_ . exact hxp (n + 1) . exact hxp (n + 2) have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) rw [mul_add, hc₂, hc₃, ← sq, ← sq] refine lt_of_sub_pos ?_ have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2 := by rw [sub_sq] linarith rw [gh₂₁] refine (sq_pos_iff).mpr ?_ refine sub_ne_zero.mpr ?_ exact hx (n+1) (n+2) (by linarith) linarith . exact ne_of_gt (hxp (n + 2)) . exact ne_of_gt (hxp (n + 1)) clear gn₀ gn₁ g₀ g₁ g₂ g₃ ga₀ ga₁ have h₃: 4 * a (n) ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) + ((x (n + 1) + x (n + 2)) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by exact imo_2023_p4_3 (fun k => x k) a hxp h₀ n hn linarith lemma imo_2023_p4_4_15 (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₀ : 0 ≤ a n + 2) -- (g₁ : 0 ≤ a (n + 2)) -- (g₂ : 0 ≤ (a n + 2) ^ 2) -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₁ : a (n + 2) = -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) (ga₀ : 1 ≤ n + 2) (ga₁ : 1 ≤ n + 1) (ga₂ : a n ^ 2 + (2 * a n * 2 + 2 ^ 2) < a n ^ 2 + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) + (x (n + 1) + x (n + 2)) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + (1 / x (n + 1) + 1 / x (n + 2)))) : a n ^ 2 + 2 * a n * 2 + 2 ^ 2 < (Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k := by rw [Finset.sum_Ico_succ_top ga₀ _, Finset.sum_Ico_succ_top ga₀ _] rw [Finset.sum_Ico_succ_top ga₁ _, Finset.sum_Ico_succ_top ga₁ _] rw [add_assoc, add_assoc, add_assoc] rw [add_mul, mul_add] rw [← gn₀] repeat rw [add_assoc] refine add_lt_add_left ?_ (a (n) ^ 2) linarith lemma imo_2023_p4_4_16 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) (g₀ : 0 ≤ a n + 2) (g₁ : 0 ≤ a (n + 2)) (g₂ : 0 ≤ (a n + 2) ^ 2) (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) (gn₁ : a (n + 2) = √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) (ga₀ : 1 ≤ n + 2) (ga₁ : 1 ≤ n + 1) : a n ^ 2 + (2 * a n * 2 + 2 ^ 2) < a n ^ 2 + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) + (x (n + 1) + x (n + 2)) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + (1 / x (n + 1) + 1 / x (n + 2))) := by repeat rw [add_assoc] refine add_lt_add_left ?_ (a (n) ^ 2) rw [mul_add (x (n + 1) + x (n + 2))] have h₂: 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)) := by repeat rw [add_mul, mul_add, mul_add] repeat rw [mul_div_left_comm _ 1 _, one_mul] repeat rw [div_self ?_] . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1) := by rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 2)) have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2) := by rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 1)) have h₂₀: 0 < x (n + 1) * x (n + 2) := by refine mul_pos ?_ ?_ . exact hxp (n + 1) . exact hxp (n + 2) have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) rw [mul_add, hc₂, hc₃, ← sq, ← sq] refine lt_of_sub_pos ?_ have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2 := by rw [sub_sq] linarith rw [gh₂₁] refine (sq_pos_iff).mpr ?_ refine sub_ne_zero.mpr ?_ exact hx (n+1) (n+2) (by linarith) linarith . exact ne_of_gt (hxp (n + 2)) . exact ne_of_gt (hxp (n + 1)) clear gn₀ gn₁ g₀ g₁ g₂ g₃ ga₀ ga₁ have h₃: 4 * a (n) ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) + ((x (n + 1) + x (n + 2)) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by exact imo_2023_p4_3 (fun k => x k) a hxp h₀ n hn linarith lemma imo_2023_p4_4_17 (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₀ : 0 ≤ a n + 2) -- (g₁ : 0 ≤ a (n + 2)) -- (g₂ : 0 ≤ (a n + 2) ^ 2) -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₁ : a (n + 2) = -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) -- (ga₀ : 1 ≤ n + 2) -- (ga₁ : 1 ≤ n + 1) (ga₂ : 2 * a n * 2 + 2 ^ 2 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) + (((x (n + 1) + x (n + 2)) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)))) : a n ^ 2 + 2 * a n * 2 + 2 ^ 2 < (Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k := by have ga₀: 1 ≤ n + 2 := by linarith rw [Finset.sum_Ico_succ_top ga₀ _, Finset.sum_Ico_succ_top ga₀ _] have ga₁: 1 ≤ n + 1 := by linarith rw [Finset.sum_Ico_succ_top ga₁ _, Finset.sum_Ico_succ_top ga₁ _] rw [add_assoc, add_assoc, add_assoc] rw [add_mul, mul_add] rw [← gn₀] repeat rw [add_assoc] refine add_lt_add_left ?_ (a (n) ^ 2) linarith lemma imo_2023_p4_4_18 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) : -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₀ : 0 ≤ a n + 2) -- (g₁ : 0 ≤ a (n + 2)) -- (g₂ : 0 ≤ (a n + 2) ^ 2) -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₁ : a (n + 2) = -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) -- (ga₀ : 1 ≤ n + 2) -- (ga₁ : 1 ≤ n + 1) : 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)) := by repeat rw [add_mul, mul_add, mul_add] repeat rw [mul_div_left_comm _ 1 _, one_mul] repeat rw [div_self ?_] . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1) := by rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 2)) have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2) := by rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 1)) have h₂₀: 0 < x (n + 1) * x (n + 2) := by refine mul_pos ?_ ?_ . exact hxp (n + 1) . exact hxp (n + 2) have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) rw [mul_add, hc₂, hc₃, ← sq, ← sq] refine lt_of_sub_pos ?_ have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2 := by rw [sub_sq] linarith rw [gh₂₁] refine (sq_pos_iff).mpr ?_ refine sub_ne_zero.mpr ?_ exact hx (n+1) (n+2) (by linarith) linarith . exact ne_of_gt (hxp (n + 2)) . exact ne_of_gt (hxp (n + 1)) lemma imo_2023_p4_4_19 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) : -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₀ : 0 ≤ a n + 2) -- (g₁ : 0 ≤ a (n + 2)) -- (g₂ : 0 ≤ (a n + 2) ^ 2) -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₁ : a (n + 2) = -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) -- (ga₀ : 1 ≤ n + 2) -- (ga₁ : 1 ≤ n + 1) -- (ga₂: 4 < x (n + 1) / x (n + 1) + x (n + 1) / x (n + 2) + (x (n + 2) / x (n + 1) + x (n + 2) / x (n + 2))) : 4 < x (n + 1) / x (n + 1) + x (n + 1) / x (n + 2) + (x (n + 2) / x (n + 1) + x (n + 2) / x (n + 2)) := by repeat rw [div_self ?_] . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1) := by rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 2)) have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2) := by rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 1)) have h₂₀: 0 < x (n + 1) * x (n + 2) := by refine mul_pos ?_ ?_ . exact hxp (n + 1) . exact hxp (n + 2) have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) rw [mul_add, hc₂, hc₃, ← sq, ← sq] refine lt_of_sub_pos ?_ have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2 := by rw [sub_sq] linarith rw [gh₂₁] refine (sq_pos_iff).mpr ?_ refine sub_ne_zero.mpr ?_ exact hx (n+1) (n+2) (by linarith) linarith . exact ne_of_gt (hxp (n + 2)) . exact ne_of_gt (hxp (n + 1)) lemma imo_2023_p4_4_20 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) : -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₀ : 0 ≤ a n + 2) -- (g₁ : 0 ≤ a (n + 2)) -- (g₂ : 0 ≤ (a n + 2) ^ 2) -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₁ : a (n + 2) = -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) -- (ga₀ : 1 ≤ n + 2) -- (ga₁ : 1 ≤ n + 1) : 4 < x (n + 1) / x (n + 1) + x (n + 1) / x (n + 2) + (x (n + 2) / x (n + 1) + x (n + 2) / x (n + 2)) := by -- repeat rw [mul_div_left_comm _ 1 _, one_mul] repeat rw [div_self ?_] . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1) := by rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 2)) have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2) := by rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 1)) have h₂₀: 0 < x (n + 1) * x (n + 2) := by refine mul_pos ?_ ?_ . exact hxp (n + 1) . exact hxp (n + 2) have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) rw [mul_add, hc₂, hc₃, ← sq, ← sq] refine lt_of_sub_pos ?_ have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2 := by rw [sub_sq] linarith rw [gh₂₁] refine (sq_pos_iff).mpr ?_ refine sub_ne_zero.mpr ?_ exact hx (n+1) (n+2) (by linarith) linarith . exact ne_of_gt (hxp (n + 2)) . exact ne_of_gt (hxp (n + 1)) lemma imo_2023_p4_4_21 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) : -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₀ : 0 ≤ a n + 2) -- (g₁ : 0 ≤ a (n + 2)) -- (g₂ : 0 ≤ (a n + 2) ^ 2) -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₁ : a (n + 2) = -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) -- (ga₀ : 1 ≤ n + 2) -- (ga₁ : 1 ≤ n + 1) : x (n + 2) ≠ 0 := by exact ne_of_gt (hxp (n + 2)) lemma imo_2023_p4_4_22 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) : -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₀ : 0 ≤ a n + 2) -- (g₁ : 0 ≤ a (n + 2)) -- (g₂ : 0 ≤ (a n + 2) ^ 2) -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₁ : a (n + 2) = -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) -- (ga₀ : 1 ≤ n + 2) -- (ga₁ : 1 ≤ n + 1) : 4 < 1 + x (n + 1) / x (n + 2) + (x (n + 2) / x (n + 1) + 1) := by have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1) := by rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 2)) have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2) := by rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 1)) have h₂₀: 0 < x (n + 1) * x (n + 2) := by refine mul_pos ?_ ?_ . exact hxp (n + 1) . exact hxp (n + 2) have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) rw [mul_add, hc₂, hc₃, ← sq, ← sq] refine lt_of_sub_pos ?_ have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2 := by rw [sub_sq] linarith rw [gh₂₁] refine (sq_pos_iff).mpr ?_ refine sub_ne_zero.mpr ?_ exact hx (n+1) (n+2) (by linarith) linarith lemma imo_2023_p4_4_23 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) : -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₀ : 0 ≤ a n + 2) -- (g₁ : 0 ≤ a (n + 2)) -- (g₂ : 0 ≤ (a n + 2) ^ 2) -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₁ : a (n + 2) = -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) -- (ga₀ : 1 ≤ n + 2) -- (ga₁ : 1 ≤ n + 1) : x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1) := by rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 2)) lemma imo_2023_p4_4_24 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) : -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₀ : 0 ≤ a n + 2) -- (g₁ : 0 ≤ a (n + 2)) -- (g₂ : 0 ≤ (a n + 2) ^ 2) -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₁ : a (n + 2) = -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) -- (ga₀ : 1 ≤ n + 2) -- (ga₁ : 1 ≤ n + 1) -- (hc₂ : x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1)) : x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2) := by have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2) := by rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 1)) linarith lemma imo_2023_p4_4_25 (x : ℕ → ℝ) -- (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₀ : 0 ≤ a n + 2) -- (g₁ : 0 ≤ a (n + 2)) -- (g₂ : 0 ≤ (a n + 2) ^ 2) -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₁ : a (n + 2) = -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) -- (ga₀ : 1 ≤ n + 2) -- (ga₁ : 1 ≤ n + 1) (hc₂ : x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1)) (hc₃ : x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2)) (h₂₀ : 0 < x (n + 1) * x (n + 2)) : 4 < 1 + x (n + 1) / x (n + 2) + (x (n + 2) / x (n + 1) + 1) := by have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) rw [mul_add, hc₂, hc₃, ← sq, ← sq] refine lt_of_sub_pos ?_ have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2 := by rw [sub_sq] linarith rw [gh₂₁] refine (sq_pos_iff).mpr ?_ refine sub_ne_zero.mpr ?_ exact hx (n+1) (n+2) (by linarith) linarith lemma imo_2023_p4_4_26 (x : ℕ → ℝ) -- (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₀ : 0 ≤ a n + 2) -- (g₁ : 0 ≤ a (n + 2)) -- (g₂ : 0 ≤ (a n + 2) ^ 2) -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₁ : a (n + 2) = -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) -- (ga₀ : 1 ≤ n + 2) -- (ga₁ : 1 ≤ n + 1) (hc₂ : x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1)) (hc₃ : x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2)) (h₂₀ : 0 < x (n + 1) * x (n + 2)) : 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) rw [mul_add, hc₂, hc₃, ← sq, ← sq] refine lt_of_sub_pos ?_ have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2 := by rw [sub_sq] linarith rw [gh₂₁] refine (sq_pos_iff).mpr ?_ refine sub_ne_zero.mpr ?_ exact hx (n+1) (n+2) (by linarith) lemma imo_2023_p4_4_27 (x : ℕ → ℝ) -- (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₀ : 0 ≤ a n + 2) -- (g₁ : 0 ≤ a (n + 2)) -- (g₂ : 0 ≤ (a n + 2) ^ 2) -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₁ : a (n + 2) = -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) -- (ga₀ : 1 ≤ n + 2) -- (ga₁ : 1 ≤ n + 1) (hc₂ : x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1)) (hc₃ : x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2)) : -- (h₂₀ : 0 < x (n + 1) * x (n + 2)) : x (n + 1) * x (n + 2) * 2 < x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1)) := by rw [mul_add, hc₂, hc₃, ← sq, ← sq] refine lt_of_sub_pos ?_ have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2 := by rw [sub_sq] linarith rw [gh₂₁] refine (sq_pos_iff).mpr ?_ refine sub_ne_zero.mpr ?_ exact hx (n+1) (n+2) (by linarith) lemma imo_2023_p4_4_28 (x : ℕ → ℝ) -- (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₀ : 0 ≤ a n + 2) -- (g₁ : 0 ≤ a (n + 2)) -- (g₂ : 0 ≤ (a n + 2) ^ 2) -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₁ : a (n + 2) = -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) -- (ga₀ : 1 ≤ n + 2) -- (ga₁ : 1 ≤ n + 1) (hc₂ : x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1)) (hc₃ : x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2)) (h₂₀ : 0 < x (n + 1) * x (n + 2)) : 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) rw [mul_add, hc₂, hc₃, ← sq, ← sq] refine lt_of_sub_pos ?_ have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2 := by rw [sub_sq] linarith rw [gh₂₁] refine (sq_pos_iff).mpr ?_ refine sub_ne_zero.mpr ?_ exact hx (n+1) (n+2) (by linarith) lemma imo_2023_p4_4_29 (x : ℕ → ℝ) -- (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) : -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₀ : 0 ≤ a n + 2) -- (g₁ : 0 ≤ a (n + 2)) -- (g₂ : 0 ≤ (a n + 2) ^ 2) -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₁ : a (n + 2) = -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) -- (ga₀ : 1 ≤ n + 2) -- (ga₁ : 1 ≤ n + 1) -- (hc₂ : x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1)) -- (hc₃ : x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2)) -- (h₂₀ : 0 < x (n + 1) * x (n + 2)) : x (n + 1) * x (n + 2) * 2 < x (n + 1) ^ 2 + x (n + 2) ^ 2 := by refine lt_of_sub_pos ?_ have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2 := by rw [sub_sq] linarith rw [gh₂₁] refine (sq_pos_iff).mpr ?_ refine sub_ne_zero.mpr ?_ exact hx (n+1) (n+2) (by linarith) lemma imo_2023_p4_4_30 (x : ℕ → ℝ) -- (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) : -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₀ : 0 ≤ a n + 2) -- (g₁ : 0 ≤ a (n + 2)) -- (g₂ : 0 ≤ (a n + 2) ^ 2) -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₁ : a (n + 2) = -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) -- (ga₀ : 1 ≤ n + 2) -- (ga₁ : 1 ≤ n + 1) -- (hc₂ : x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1)) -- (hc₃ : x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2)) -- (h₂₀ : 0 < x (n + 1) * x (n + 2)) : 0 < x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 := by have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2 := by rw [sub_sq] linarith rw [gh₂₁] refine (sq_pos_iff).mpr ?_ refine sub_ne_zero.mpr ?_ exact hx (n+1) (n+2) (by linarith) lemma imo_2023_p4_4_31 (x : ℕ → ℝ) -- (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) : -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₀ : 0 ≤ a n + 2) -- (g₁ : 0 ≤ a (n + 2)) -- (g₂ : 0 ≤ (a n + 2) ^ 2) -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₁ : a (n + 2) = -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) -- (ga₀ : 1 ≤ n + 2) -- (ga₁ : 1 ≤ n + 1) -- (hc₂ : x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1)) -- (hc₃ : x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2)) -- (h₂₀ : 0 < x (n + 1) * x (n + 2)) : x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2 := by rw [sub_sq] linarith lemma imo_2023_p4_4_32 (x : ℕ → ℝ) -- (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₀ : 0 ≤ a n + 2) -- (g₁ : 0 ≤ a (n + 2)) -- (g₂ : 0 ≤ (a n + 2) ^ 2) -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₁ : a (n + 2) = -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) -- (ga₀ : 1 ≤ n + 2) -- (ga₁ : 1 ≤ n + 1) -- (hc₂ : x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1)) -- (hc₃ : x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2)) -- (h₂₀ : 0 < x (n + 1) * x (n + 2)) (gh₂₁ : x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2) : 0 < x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 := by rw [gh₂₁] refine (sq_pos_iff).mpr ?_ refine sub_ne_zero.mpr ?_ exact hx (n+1) (n+2) (by linarith) lemma imo_2023_p4_4_33 (x : ℕ → ℝ) -- (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) : -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₀ : 0 ≤ a n + 2) -- (g₁ : 0 ≤ a (n + 2)) -- (g₂ : 0 ≤ (a n + 2) ^ 2) -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₁ : a (n + 2) = -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) -- (ga₀ : 1 ≤ n + 2) -- (ga₁ : 1 ≤ n + 1) -- (hc₂ : x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1)) -- (hc₃ : x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2)) -- (h₂₀ : 0 < x (n + 1) * x (n + 2)) -- (gh₂₁ : x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2) : x (n + 1) - x (n + 2) ≠ 0 := by refine sub_ne_zero.mpr ?_ exact hx (n+1) (n+2) (by linarith) lemma imo_2023_p4_4_34 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) (h₂ : 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2))) : 2 * a n * 2 + 2 ^ 2 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) + (((x (n + 1) + x (n + 2)) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2))) := by have h₃: 4 * a (n) ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) + ((x (n + 1) + x (n + 2)) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by exact imo_2023_p4_3 (fun k => x k) a hxp h₀ n hn linarith lemma imo_2023_p4_4_35 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (h₂ : 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2))) (h₃ : 4 * a n ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) + (x (n + 1) + x (n + 2)) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) : 2 * a n * 2 + 2 ^ 2 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) + (((x (n + 1) + x (n + 2)) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2))) := by have h₂: 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)) := by repeat rw [add_mul, mul_add, mul_add] repeat rw [mul_div_left_comm _ 1 _, one_mul] repeat rw [div_self ?_] . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1) := by rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 2)) have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2) := by rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] simp exact ne_of_gt (hxp (n + 1)) have h₂₀: 0 < x (n + 1) * x (n + 2) := by refine mul_pos ?_ ?_ . exact hxp (n + 1) . exact hxp (n + 2) have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) rw [mul_add, hc₂, hc₃, ← sq, ← sq] refine lt_of_sub_pos ?_ have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2 := by rw [sub_sq] linarith rw [gh₂₁] refine (sq_pos_iff).mpr ?_ refine sub_ne_zero.mpr ?_ exact hx (n+1) (n+2) (by linarith) linarith . exact ne_of_gt (hxp (n + 2)) . exact ne_of_gt (hxp (n + 1)) linarith lemma imo_2023_p4_4_36 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) : -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₀ : 0 ≤ a n + 2) -- (g₁ : 0 ≤ a (n + 2)) -- (g₂ : 0 ≤ (a n + 2) ^ 2) -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₁ : a (n + 2) = -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k := by refine mul_nonneg ?_ ?_ . refine Finset.sum_nonneg ?_ intros i _ exact LT.lt.le (hxp i) . refine Finset.sum_nonneg ?_ intros i _ simp exact LT.lt.le (hxp i) lemma imo_2023_p4_4_37 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) : -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₀ : 0 ≤ a n + 2) -- (g₁ : 0 ≤ a (n + 2)) -- (g₂ : 0 ≤ (a n + 2) ^ 2) -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₁ : a (n + 2) = -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) : 0 ≤ Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k := by refine Finset.sum_nonneg ?_ intros i _ exact LT.lt.le (hxp i) lemma imo_2023_p4_4_38 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) : -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₀ : 0 ≤ a n + 2) -- (g₁ : 0 ≤ a (n + 2)) -- (g₂ : 0 ≤ (a n + 2) ^ 2) -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) -- (gn₁ : a (n + 2) = -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) : 0 ≤ Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k := by refine Finset.sum_nonneg ?_ intros i _ simp exact LT.lt.le (hxp i) lemma imo_2023_p4_5 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (h₁ : ∀ (n : ℕ), (1 ≤ n ∧ n ≤ 2023) → ∃ (kz:ℤ), (a n = ↑kz )) (ha1 : a 1 = 1) : 3034 ≤ a 2023 := by have h₀₁: ∀ (n : ℕ), (1 ≤ n ∧ n ≤ 2023) → 0 < a n := by intros n hn have ha: a n = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by exact h₀ (n) (hn) rw [ha] refine Real.sqrt_pos.mpr ?_ refine mul_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . intros i _ exact hxp i . simp linarith . refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith have h₁₁: ∀ (n : ℕ), (1 ≤ n ∧ n ≤ 2023) → ∃ (kn:ℕ), a n = ↑kn := by intros n hn have g₁₁: 0 < a n := by exact h₀₁ n hn let ⟨p, gp⟩ := h₁ n hn let q:ℕ := Int.toNat p have g₁₂: ↑q = p := by refine Int.toNat_of_nonneg ?_ rw [gp] at g₁₁ norm_cast at g₁₁ exact Int.le_of_lt g₁₁ use q rw [gp] norm_cast exact id g₁₂.symm have h₂₁: ∀ (n:ℕ), (1 ≤ n ∧ n ≤ 2021) → a n + 2 < a (n+2) := by exact fun n a_1 => imo_2023_p4_4 (fun i => x i) a hxp hx h₀ h₀₁ n a_1 have h₂: ∀ (n:ℕ), (1 ≤ n ∧ n ≤ 2021) → a n + 3 ≤ a (n+2) := by intros n hn have g₀: a n + 2 < a (n + 2) := by exact h₂₁ n hn have g₀₁: ∃ (p:ℕ), a n = ↑p := by apply h₁₁ constructor . linarith . linarith have g₀₂: ∃ (q:ℕ), a (n + 2) = ↑q := by apply h₁₁ constructor . linarith . linarith let ⟨p, _⟩ := g₀₁ let ⟨q, _⟩ := g₀₂ have g₁: p + 2 < q := by suffices g₁₁: ↑p + (2:ℝ) < ↑q . norm_cast at g₁₁ . linarith have g₂: ↑p + (3:ℝ) ≤ ↑q := by norm_cast linarith have h₃: ∀ (n:ℕ), (0 ≤ n ∧ n ≤ 1010) → a 1 + 3 * (↑(n) + 1) ≤ a (3 + 2 * n) := by intros n hn induction' n with d hd · simp exact h₂ (1) (by norm_num) · rw [mul_add] simp have g₀: a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1)) := by refine h₂ (3 + 2 * d) ?_ constructor . linarith . linarith have g₁: a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3 := by refine add_le_add_right ?_ (3) apply hd constructor . linarith . linarith refine le_trans (by linarith[g₁]) g₀ rw [ha1] at h₃ have h₄: (3034:ℝ) = 1 + 3 * (↑1010 + 1) := by norm_num rw [h₄] exact h₃ (1010) (by norm_num) lemma imo_2023_p4_5_1 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) : -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ kz, a n = kz) -- (ha1 : a 1 = 1) : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n := by intros n hn have ha: a n = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by exact h₀ (n) (hn) rw [ha] refine Real.sqrt_pos.mpr ?_ refine mul_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . intros i _ exact hxp i . simp linarith . refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith lemma imo_2023_p4_5_2 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ kz, a n = kz) -- (ha1 : a 1 = 1) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2023) (ha : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) : 0 < a n := by rw [ha] refine Real.sqrt_pos.mpr ?_ refine mul_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . intros i _ exact hxp i . simp linarith . refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith lemma imo_2023_p4_5_3 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ kz, a n = kz) -- (ha1 : a 1 = 1) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2023) : -- (ha : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) : 0 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k := by refine mul_pos ?_ ?_ . refine Finset.sum_pos ?_ ?_ . intros i _ exact hxp i . simp linarith . refine Finset.sum_pos ?_ ?_ . intros i _ exact one_div_pos.mpr (hxp i) . simp linarith lemma imo_2023_p4_5_4 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = kz) (ha1 : a 1 = 1) (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) : 3034 ≤ a 2023 := by have h₁₁: ∀ (n : ℕ), (1 ≤ n ∧ n ≤ 2023) → ∃ (kn:ℕ), a n = ↑kn := by intros n hn have g₁₁: 0 < a n := by exact h₀₁ n hn let ⟨p, gp⟩ := h₁ n hn let q:ℕ := Int.toNat p have g₁₂: ↑q = p := by refine Int.toNat_of_nonneg ?_ rw [gp] at g₁₁ norm_cast at g₁₁ exact Int.le_of_lt g₁₁ use q rw [gp] norm_cast exact id g₁₂.symm have h₂₁: ∀ (n:ℕ), (1 ≤ n ∧ n ≤ 2021) → a n + 2 < a (n+2) := by exact fun n a_1 => imo_2023_p4_4 (fun i => x i) a hxp hx h₀ h₀₁ n a_1 have h₂: ∀ (n:ℕ), (1 ≤ n ∧ n ≤ 2021) → a n + 3 ≤ a (n+2) := by intros n hn have g₀: a n + 2 < a (n + 2) := by exact h₂₁ n hn have g₀₁: ∃ (p:ℕ), a n = ↑p := by apply h₁₁ constructor . linarith . linarith have g₀₂: ∃ (q:ℕ), a (n + 2) = ↑q := by apply h₁₁ constructor . linarith . linarith let ⟨p, _⟩ := g₀₁ let ⟨q, _⟩ := g₀₂ have g₁: p + 2 < q := by suffices g₁₁: ↑p + (2:ℝ) < ↑q . norm_cast at g₁₁ . linarith have g₂: ↑p + (3:ℝ) ≤ ↑q := by norm_cast linarith have h₃: ∀ (n:ℕ), (0 ≤ n ∧ n ≤ 1010) → a 1 + 3 * (↑(n) + 1) ≤ a (3 + 2 * n) := by intros n hn induction' n with d hd · simp exact h₂ (1) (by norm_num) · rw [mul_add] simp have g₀: a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1)) := by refine h₂ (3 + 2 * d) ?_ constructor . linarith . linarith have g₁: a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3 := by refine add_le_add_right ?_ (3) apply hd constructor . linarith . linarith refine le_trans (by linarith[g₁]) g₀ rw [ha1] at h₃ have h₄: (3034:ℝ) = 1 + 3 * (↑1010 + 1) := by norm_num rw [h₄] exact h₃ (1010) (by norm_num) lemma imo_2023_p4_5_5 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = kz) -- (ha1 : a 1 = 1) (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn := by intros n hn have g₁₁: 0 < a n := by exact h₀₁ n hn let ⟨p, gp⟩ := h₁ n hn let q:ℕ := Int.toNat p have g₁₂: ↑q = p := by refine Int.toNat_of_nonneg ?_ rw [gp] at g₁₁ norm_cast at g₁₁ exact Int.le_of_lt g₁₁ use q rw [gp] norm_cast exact id g₁₂.symm lemma imo_2023_p4_5_6 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = kz) -- (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2023) (g₁₁ : 0 < a n) : ∃ (kn:ℕ), a n = ↑kn := by let ⟨p, gp⟩ := h₁ n hn let q:ℕ := Int.toNat p have g₁₂: ↑q = p := by refine Int.toNat_of_nonneg ?_ rw [gp] at g₁₁ norm_cast at g₁₁ exact Int.le_of_lt g₁₁ use q rw [gp] norm_cast exact id g₁₂.symm lemma imo_2023_p4_5_7 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) -- (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n q : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2023) (g₁₁ : 0 < a n) (p : ℤ) (gp : a n = ↑p) (hq : q = Int.toNat p) : ∃ kn:ℕ, a n = ↑kn := by have g₁₂: (↑q:ℤ) = p := by rw [hq] refine Int.toNat_of_nonneg ?_ rw [gp] at g₁₁ norm_cast at g₁₁ exact Int.le_of_lt g₁₁ use q rw [gp] exact congrArg Int.cast (id g₁₂.symm) lemma imo_2023_p4_5_8 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) -- (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n q : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2023) (g₁₁ : 0 < a n) (p : ℤ) (gp : a n = ↑p) (hq : q = Int.toNat p) : ↑q = p := by rw [hq] refine Int.toNat_of_nonneg ?_ rw [gp] at g₁₁ norm_cast at g₁₁ exact Int.le_of_lt g₁₁ lemma imo_2023_p4_5_9 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) -- (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n q : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2023) (g₁₁ : 0 < a n) (p : ℤ) (gp : a n = ↑p) (hq : q = Int.toNat p) : 0 ≤ p := by rw [gp] at g₁₁ norm_cast at g₁₁ exact Int.le_of_lt g₁₁ lemma imo_2023_p4_5_10 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) -- (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n q : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2023) -- (g₁₁ : 0 < a n) (p : ℤ) (gp : a n = ↑p) -- (hq : q = Int.toNat p) (g₁₂ : ↑q = p) : ∃ (kn:ℕ), a n = ↑kn := by use q rw [gp] norm_cast exact id g₁₂.symm lemma imo_2023_p4_5_11 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) -- (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (n : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2023) -- (g₁₁ : 0 < a n) (p : ℤ) (gp : a n = ↑p) (q : ℕ := Int.toNat p) (g₁₂ : ↑q = p) : a n = ↑q := by rw [gp] norm_cast exact id g₁₂.symm lemma imo_2023_p4_5_12 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) (ha1 : a 1 = 1) (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) : 3034 ≤ a 2023 := by have h₂₁: ∀ (n:ℕ), (1 ≤ n ∧ n ≤ 2021) → a n + 2 < a (n+2) := by exact fun n a_1 => imo_2023_p4_4 (fun i => x i) a hxp hx h₀ h₀₁ n a_1 have h₂: ∀ (n:ℕ), (1 ≤ n ∧ n ≤ 2021) → a n + 3 ≤ a (n+2) := by intros n hn have g₀: a n + 2 < a (n + 2) := by exact h₂₁ n hn have g₀₁: ∃ (p:ℕ), a n = ↑p := by apply h₁₁ constructor . linarith . linarith have g₀₂: ∃ (q:ℕ), a (n + 2) = ↑q := by apply h₁₁ constructor . linarith . linarith let ⟨p, _⟩ := g₀₁ let ⟨q, _⟩ := g₀₂ have g₁: p + 2 < q := by suffices g₁₁: ↑p + (2:ℝ) < ↑q . norm_cast at g₁₁ . linarith have g₂: ↑p + (3:ℝ) ≤ ↑q := by norm_cast linarith have h₃: ∀ (n:ℕ), (0 ≤ n ∧ n ≤ 1010) → a 1 + 3 * (↑(n) + 1) ≤ a (3 + 2 * n) := by intros n hn induction' n with d hd · simp exact h₂ (1) (by norm_num) · rw [mul_add] simp have g₀: a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1)) := by refine h₂ (3 + 2 * d) ?_ constructor . linarith . linarith have g₁: a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3 := by refine add_le_add_right ?_ (3) apply hd constructor . linarith . linarith refine le_trans (by linarith[g₁]) g₀ rw [ha1] at h₃ have h₄: (3034:ℝ) = 1 + 3 * (↑1010 + 1) := by norm_num rw [h₄] exact h₃ (1010) (by norm_num) lemma imo_2023_p4_5_13 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) : 3034 ≤ a 2023 := by have h₂: ∀ (n:ℕ), (1 ≤ n ∧ n ≤ 2021) → a n + 3 ≤ a (n+2) := by intros n hn have g₀: a n + 2 < a (n + 2) := by exact h₂₁ n hn have g₀₁: ∃ (p:ℕ), a n = ↑p := by apply h₁₁ constructor . linarith . linarith have g₀₂: ∃ (q:ℕ), a (n + 2) = ↑q := by apply h₁₁ constructor . linarith . linarith let ⟨p, _⟩ := g₀₁ let ⟨q, _⟩ := g₀₂ have g₁: p + 2 < q := by suffices g₁₁: ↑p + (2:ℝ) < ↑q . norm_cast at g₁₁ . linarith have g₂: ↑p + (3:ℝ) ≤ ↑q := by norm_cast linarith have h₃: ∀ (n:ℕ), (0 ≤ n ∧ n ≤ 1010) → a 1 + 3 * (↑(n) + 1) ≤ a (3 + 2 * n) := by intros n hn induction' n with d hd · simp exact h₂ (1) (by norm_num) · rw [mul_add] simp have g₀: a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1)) := by refine h₂ (3 + 2 * d) ?_ constructor . linarith . linarith have g₁: a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3 := by refine add_le_add_right ?_ (3) apply hd constructor . linarith . linarith refine le_trans (by linarith[g₁]) g₀ rw [ha1] at h₃ have h₄: (3034:ℝ) = 1 + 3 * (↑1010 + 1) := by norm_num rw [h₄] exact h₃ (1010) (by norm_num) lemma imo_2023_p4_5_14 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) -- (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2) := by intros n hn have g₀: a n + 2 < a (n + 2) := by exact h₂₁ n hn have g₀₁: ∃ (p:ℕ), a n = ↑p := by apply h₁₁ constructor . linarith . linarith have g₀₂: ∃ (q:ℕ), a (n + 2) = ↑q := by apply h₁₁ constructor . linarith . linarith let ⟨p, _⟩ := g₀₁ let ⟨q, _⟩ := g₀₂ have g₁: p + 2 < q := by suffices g₁₁: ↑p + (2:ℝ) < ↑q . norm_cast at g₁₁ . linarith have g₂: ↑p + (3:ℝ) ≤ ↑q := by norm_cast linarith lemma imo_2023_p4_5_15 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) -- (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) (g₀ : a n + 2 < a (n + 2)) : a n + 3 ≤ a (n + 2) := by have g₀₁: ∃ (p:ℕ), a n = ↑p := by apply h₁₁ constructor . linarith . linarith have g₀₂: ∃ (q:ℕ), a (n + 2) = ↑q := by apply h₁₁ constructor . linarith . linarith let ⟨p, _⟩ := g₀₁ let ⟨q, _⟩ := g₀₂ have g₁: p + 2 < q := by suffices g₁₁: ↑p + (2:ℝ) < ↑q . norm_cast at g₁₁ . linarith have g₂: ↑p + (3:ℝ) ≤ ↑q := by norm_cast linarith lemma imo_2023_p4_5_16 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) -- (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) : -- (g₀ : a n + 2 < a (n + 2)) : ∃ (p:ℕ), a n = ↑p := by apply h₁₁ constructor . linarith . linarith lemma imo_2023_p4_5_17 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) -- (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) (n : ℕ) (hn : 1 ≤ n ∧ n ≤ 2021) (g₀ : a n + 2 < a (n + 2)) (g₀₁ : ∃ (p:ℕ), a n = ↑p) : a n + 3 ≤ a (n + 2) := by have g₀₂: ∃ (q:ℕ), a (n + 2) = ↑q := by apply h₁₁ constructor . linarith . linarith let ⟨p, _⟩ := g₀₁ let ⟨q, _⟩ := g₀₂ have g₁: p + 2 < q := by suffices g₁₁: ↑p + (2:ℝ) < ↑q . norm_cast at g₁₁ . linarith have g₂: ↑p + (3:ℝ) ≤ ↑q := by norm_cast linarith lemma imo_2023_p4_5_18 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) -- (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) (n : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2021) (g₀ : a n + 2 < a (n + 2)) (g₀₁ : ∃ (p:ℕ), a n = ↑p) (g₀₂ : ∃ (q:ℕ), a (n + 2) = ↑q) : a n + 3 ≤ a (n + 2) := by let ⟨p, _⟩ := g₀₁ let ⟨q, _⟩ := g₀₂ have g₁: p + 2 < q := by suffices g₁₁: ↑p + (2:ℝ) < ↑q . norm_cast at g₁₁ . linarith have g₂: ↑p + (3:ℝ) ≤ ↑q := by norm_cast linarith lemma imo_2023_p4_5_19 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) -- (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) (n : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2021) (g₀ : a n + 2 < a (n + 2)) -- (g₀₁ : ∃ p, a n = ↑p) -- (g₀₂ : ∃ q, a (n + 2) = ↑q) (p : ℕ) (h₈ : a n = ↑p) (q : ℕ) (h₉ : a (n + 2) = ↑q) : a n + 3 ≤ a (n + 2) := by have g₁: p + 2 < q := by suffices g₁₁: ↑p + (2:ℝ) < ↑q . norm_cast at g₁₁ . linarith have g₂: ↑p + (3:ℝ) ≤ ↑q := by norm_cast linarith lemma imo_2023_p4_5_20 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) -- (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) (n : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2021) (g₀ : a n + 2 < a (n + 2)) -- (g₀₁ : ∃ p, a n = ↑p) -- (g₀₂ : ∃ q, a (n + 2) = ↑q) (p : ℕ) (h₈ : a n = ↑p) (q : ℕ) (h₉ : a (n + 2) = ↑q) : p + 2 < q := by suffices g₁₁: ↑p + (2:ℝ) < ↑q . norm_cast at g₁₁ . linarith lemma imo_2023_p4_5_21 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) -- (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) (n : ℕ) -- (hn : 1 ≤ n ∧ n ≤ 2021) -- (g₀ : a n + 2 < a (n + 2)) -- (g₀₁ : ∃ p, a n = ↑p) -- (g₀₂ : ∃ q, a (n + 2) = ↑q) (p : ℕ) (h₈ : a n = ↑p) (q : ℕ) (h₉ : a (n + 2) = ↑q) (g₁ : p + 2 < q) : a n + 3 ≤ a (n + 2) := by have g₂: ↑p + (3:ℝ) ≤ ↑q := by norm_cast linarith lemma imo_2023_p4_5_22 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) : 3034 ≤ a 2023 := by have h₃: ∀ (n:ℕ), (0 ≤ n ∧ n ≤ 1010) → a 1 + 3 * (↑(n) + 1) ≤ a (3 + 2 * n) := by intros n hn induction' n with d hd · simp exact h₂ (1) (by norm_num) · rw [mul_add] simp have g₀: a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1)) := by refine h₂ (3 + 2 * d) ?_ constructor . linarith . linarith have g₁: a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3 := by refine add_le_add_right ?_ (3) apply hd constructor . linarith . linarith refine le_trans (by linarith[g₁]) g₀ rw [ha1] at h₃ have h₄: (3034:ℝ) = 1 + 3 * (↑1010 + 1) := by norm_num rw [h₄] exact h₃ (1010) (by norm_num) lemma imo_2023_p4_5_23 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) -- (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) : ∀ (n : ℕ), 0 ≤ n ∧ n ≤ 1010 → a 1 + 3 * (↑n + 1) ≤ a (3 + 2 * n) := by intros n hn induction' n with d hd · simp exact h₂ (1) (by norm_num) · rw [mul_add] simp have g₀: a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1)) := by refine h₂ (3 + 2 * d) ?_ constructor . linarith . linarith have g₁: a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3 := by refine add_le_add_right ?_ (3) apply hd constructor . linarith . linarith refine le_trans (by linarith[g₁]) g₀ lemma imo_2023_p4_5_24 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) -- (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) (n : ℕ) (hn : 0 ≤ n ∧ n ≤ 1010) : a 1 + 3 * (↑n + 1) ≤ a (3 + 2 * n) := by induction' n with d hd · simp exact h₂ (1) (by norm_num) · rw [mul_add] simp have g₀: a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1)) := by refine h₂ (3 + 2 * d) ?_ constructor . linarith . linarith have g₁: a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3 := by refine add_le_add_right ?_ (3) apply hd constructor . linarith . linarith refine le_trans (by linarith[g₁]) g₀ lemma imo_2023_p4_5_25 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) -- (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) : -- (hn : 0 ≤ Nat.zero ∧ Nat.zero ≤ 1010) : a 1 + 3 * (↑Nat.zero + 1) ≤ a (3 + 2 * Nat.zero) := by simp exact h₂ (1) (by norm_num) lemma imo_2023_p4_5_26 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) -- (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) (d : ℕ) (hd : 0 ≤ d ∧ d ≤ 1010 → a 1 + 3 * (↑d + 1) ≤ a (3 + 2 * d)) (hn : 0 ≤ Nat.succ d ∧ Nat.succ d ≤ 1010) : a 1 + 3 * (↑(Nat.succ d) + 1) ≤ a (3 + 2 * Nat.succ d) := by rw [mul_add, Nat.succ_eq_add_one] simp have g₀: a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1)) := by refine h₂ (3 + 2 * d) ?_ constructor . linarith . linarith have g₁: a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3 := by refine add_le_add_right ?_ (3) apply hd constructor . linarith . linarith refine le_trans (by linarith[g₁]) g₀ lemma imo_2023_p4_5_27 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) -- (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) (d : ℕ) (hd : 0 ≤ d ∧ d ≤ 1010 → a 1 + 3 * (↑d + 1) ≤ a (3 + 2 * d)) (hn : 0 ≤ Nat.succ d ∧ Nat.succ d ≤ 1010) : a 1 + (3 * (↑d + 1) + 3) ≤ a (3 + 2 * (d + 1)) := by have g₀: a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1)) := by refine h₂ (3 + 2 * d) ?_ constructor . linarith . linarith have g₁: a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3 := by refine add_le_add_right ?_ (3) apply hd constructor . linarith . linarith refine le_trans (by linarith[g₁]) g₀ lemma imo_2023_p4_5_28 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) -- (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) (d : ℕ) -- (hd : 0 ≤ d ∧ d ≤ 1010 → a 1 + 3 * (↑d + 1) ≤ a (3 + 2 * d)) (hn : 0 ≤ Nat.succ d ∧ Nat.succ d ≤ 1010) : a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1)) := by refine h₂ (3 + 2 * d) ?_ constructor . linarith . linarith lemma imo_2023_p4_5_29 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) -- (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) -- (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) (d : ℕ) (hd : 0 ≤ d ∧ d ≤ 1010 → a 1 + 3 * (↑d + 1) ≤ a (3 + 2 * d)) (hn : 0 ≤ Nat.succ d ∧ Nat.succ d ≤ 1010) (g₀ : a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1))) : a 1 + (3 * (↑d + 1) + 3) ≤ a (3 + 2 * (d + 1)) := by have g₁: a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3 := by refine add_le_add_right ?_ (3) apply hd constructor . linarith . linarith refine le_trans (by linarith[g₁]) g₀ lemma imo_2023_p4_5_30 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) -- (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) -- (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) (d : ℕ) (hd : 0 ≤ d ∧ d ≤ 1010 → a 1 + 3 * (↑d + 1) ≤ a (3 + 2 * d)) (hn : 0 ≤ Nat.succ d ∧ Nat.succ d ≤ 1010) : -- (g₀ : a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1))) : a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3 := by refine add_le_add_right ?_ (3) apply hd constructor . linarith . linarith lemma imo_2023_p4_5_31 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) -- (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) -- (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) (d : ℕ) (hd : 0 ≤ d ∧ d ≤ 1010 → a 1 + 3 * (↑d + 1) ≤ a (3 + 2 * d)) (hn : 0 ≤ Nat.succ d ∧ Nat.succ d ≤ 1010) : -- (g₀ : a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1))) : a 1 + 3 * (↑d + 1) ≤ a (3 + 2 * d) := by apply hd constructor . linarith . linarith lemma imo_2023_p4_5_32 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) -- (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) -- (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) (d : ℕ) -- (hd : 0 ≤ d ∧ d ≤ 1010 → a 1 + 3 * (↑d + 1) ≤ a (3 + 2 * d)) -- (hn : 0 ≤ Nat.succ d ∧ Nat.succ d ≤ 1010) (g₀ : a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1))) (g₁ : a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3) : a 1 + (3 * (↑d + 1) + 3) ≤ a (3 + 2 * (d + 1)) := by exact le_trans (by linarith[g₁]) g₀ lemma imo_2023_p4_5_33 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) -- (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) (h₃ : ∀ (n : ℕ), 0 ≤ n ∧ n ≤ 1010 → a 1 + 3 * (↑n + 1) ≤ a (3 + 2 * n)) : 3034 ≤ a 2023 := by rw [ha1] at h₃ have h₄: (3034:ℝ) = 1 + 3 * (↑1010 + 1) := by norm_num rw [h₄] exact h₃ (1010) (by norm_num) lemma imo_2023_p4_5_34 -- (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) -- (ha1 : a 1 = 1) -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) -- (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) (h₃ : ∀ (n : ℕ), 0 ≤ n ∧ n ≤ 1010 → 1 + 3 * (↑n + 1) ≤ a (3 + 2 * n)) (h₄ : (3034:ℝ) = 1 + 3 * (↑1010 + 1)) : 3034 ≤ a 2023 := by rw [h₄] exact h₃ (1010) (by norm_num) lemma imo_2023_p4_6 (x : ℕ → ℝ) (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) : -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) : a 1 = 1 := by have g₀: sqrt ((Finset.sum (Finset.Ico 1 (1 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (1 + 1)) fun k => 1 / x k) = 1 := by norm_num refine div_self ?_ exact ne_of_gt (hxp 1) rw [← g₀] exact h₀ (1) (by norm_num) lemma imo_2023_p4_6_1 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) : -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * -- Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) : -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) : √((Finset.sum (Finset.Ico 1 (1 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (1 + 1)) fun k => 1 / x k) = 1 := by norm_num refine div_self ?_ exact ne_of_gt (hxp 1) lemma imo_2023_p4_6_2 (x : ℕ → ℝ) -- (a : ℕ → ℝ) (hxp : ∀ (i : ℕ), 0 < x i) : -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) -- (h₀ : ∀ (n : ℕ), -- 1 ≤ n ∧ n ≤ 2023 → -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) : x 1 * (x 1)⁻¹ = 1 := by refine div_self ?_ exact ne_of_gt (hxp 1) lemma imo_2023_p4_6_3 (x : ℕ → ℝ) (a : ℕ → ℝ) -- (hxp : ∀ (i : ℕ), 0 < x i) -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) (h₀ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) (g₀ : √((Finset.sum (Finset.Ico 1 (1 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (1 + 1)) fun k => 1 / x k) = 1) : a 1 = 1 := by rw [← g₀] exact h₀ (1) (by norm_num)