import Mathlib set_option linter.unusedVariables.analyzeTactics true open NNReal Nat BigOperators Finset -- imo-official.org/problems/IMO2007SL.pdf lemma aux1 (a : ℕ → NNReal) (m : ℕ) (hm₀ : Nat.succ 4 ≤ m) : a (m - 1) ^ 2 + a m ^ 2 + a 1 ^ 2 + a 2 ^ 2 ≤ ∑ x ∈ Finset.range m, a (x + 1) ^ 2 := by let fs: Finset ℕ := {0, 1, m-2, m-1} have h₀: fs = {0, 1, m-2, m-1} := by rfl have h₁: fs ⊆ Finset.range m := by refine insert_subset ?_ ?_ . refine mem_range.mpr ?_ exact zero_lt_of_lt hm₀ . refine insert_subset ?_ ?_ . refine mem_range.mpr ?_ linarith . refine insert_subset ?_ ?_ . refine mem_range.mpr ?_ refine sub_lt ?_ (by norm_num) exact zero_lt_of_lt hm₀ . refine singleton_subset_iff.mpr ?_ refine mem_range.mpr ?_ exact sub_one_lt_of_lt hm₀ rw [← Finset.sum_sdiff h₁] have h₂: ∑ x ∈ fs, a (x + 1) ^ 2 = a (m - 1) ^ 2 + a m ^ 2 + a 1 ^ 2 + a 2 ^ 2 := by rw [h₀] have g₀: 0 ∈ fs := by exact mem_insert_self 0 {1, m - 2, m - 1} rw [← Finset.add_sum_erase fs _ g₀] simp have g₁: 4 ≤ m - 1 := by exact Nat.le_sub_one_of_lt hm₀ have g₂: 3 ≤ m - 2 := by exact le_sub_of_add_le hm₀ have g₃: fs.erase 0 = ({1, m - 2, m - 1}:(Finset ℕ)) := by rw [h₀] refine erase_insert ?h refine forall_mem_not_eq'.mp ?_ intros b hb₀ hb₁ rw [hb₁] at hb₀ norm_num at hb₀ cases' hb₀ with hb₀ hb₀ . rw [← hb₀] at g₂ linarith . rw [← hb₀] at g₁ linarith rw [g₃] have g₄: (1:ℕ) ∈ ({1, m - 2, m - 1}:(Finset ℕ)) := by exact mem_insert_self 1 {m - 2, m - 1} rw [← Finset.add_sum_erase _ _ g₄] simp rw [Finset.erase_eq_self.mpr ?_] . have g₅: (m - 2) ∈ ({m - 2, m - 1}:(Finset ℕ)) := by exact mem_insert_self (m - 2) {m - 1} rw [← Finset.add_sum_erase _ _ g₅] simp rw [Finset.erase_eq_self.mpr ?_] . rw [Finset.sum_singleton, Nat.sub_add_cancel (by linarith)] rw [← Nat.sub_add_comm (by linarith)] simp ring_nf . refine Finset.not_mem_singleton.mpr ?_ omega . refine forall_mem_not_eq'.mp ?_ intros b hb₀ hb₁ rw [hb₁] at hb₀ simp at hb₀ cases' hb₀ with hb₀ hb₀ . rw [← hb₀] at g₂ linarith . rw [← hb₀] at g₁ linarith rw [add_comm _ (∑ x ∈ fs, a (x + 1) ^ 2), h₂] exact le_self_add lemma aux2 (a : ℕ → NNReal) : ∀ (n : ℕ), 4 < n ∧ n < 101 → (∀ (x y : ℕ), x % n = y % n → a (x + 1) = a (y + 1)) → ∑ x ∈ range n, (a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) ≤ (∑ x ∈ range n, a (x + 1) ^ 2) ^ 2 := by intro n hn₀ hn₂ cases' hn₀ with hn₀ hn₁ have hn₃: n = (n - 2) + 1 + 1 := by omega nth_rw 1 [hn₃,] rw [Finset.sum_range_succ, sum_range_succ] have hn₄: a (n - 2 + 1) = a (n - 1) := by refine congrArg a (by omega) have hn₅: a (n - 2 + 3) = a 1 := by refine hn₂ (n - 2 + 2) 0 ?_ rw [Nat.zero_mod, Nat.sub_add_cancel ?_] . rw [Nat.mod_self n] . linarith have hn₆: a (n - 2 + 1 + 3) = a 2 := by refine hn₂ (n - 2 + 3) 1 ?_ symm rw [Nat.mod_eq_of_lt (by linarith)] have g₀: n - 2 + 3 = n + 1 := by linarith rw [g₀] refine Eq.symm (mod_eq_of_modEq ?_ (by linarith)) exact Nat.add_modEq_left rw [← hn₃, hn₄, hn₅, hn₆] refine le_induction ?_ ?_ n hn₀ . repeat rw [Finset.sum_range_succ] simp ring_nf repeat refine add_le_add_right ?_ _ refine le_of_eq ?_ rfl . intros m hm₀ hm₁ have hm₂: m + 1 - 2 = m - 2 + 1 := by rw [add_comm, add_comm _ 1, Nat.add_sub_assoc ?_ 1] omega rw [hm₂, Finset.sum_range_succ, sum_range_succ] have hm₃: m - 2 + 1 = m - 1 := by exact id (Eq.symm hm₂) have hm₄: m - 2 + 2 = m := by exact Eq.symm ((fun {m n} => pred_eq_succ_iff.mp) hm₂) have hm₅: m - 2 + 3 = m + 1 := by omega have hm₆: m + 1 - 1 = m := by exact rfl rw [hm₃, hm₄, hm₅, hm₆] clear hm₃ hm₄ hm₅ hm₆ rw [add_sq, add_assoc ((∑ x ∈ Finset.range m, a (x + 1) ^ 2) ^ 2)] have h₅₀: 2 * a (m - 1) ^ 2 * a (m + 1) ^ 2 + 2 * a m ^ 2 * a (m + 1) ^ 2 + 2 * a (m + 1) ^ 2 * a 1 ^ 2 + 2 * a (m + 1) ^ 2 * a 2 ^ 2 + a (m + 1) ^ 4 ≤ (2 * ∑ x ∈ Finset.range m, a (x + 1) ^ 2) * a (m + 1) ^ 2 + (a (m + 1) ^ 2) ^ 2 := by rw [← pow_mul] simp have h₅₁: 2 * a (m - 1) ^ 2 * a (m + 1) ^ 2 + 2 * a m ^ 2 * a (m + 1) ^ 2 + 2 * a (m + 1) ^ 2 * a 1 ^ 2 + 2 * a (m + 1) ^ 2 * a 2 ^ 2 = 2 * a (m + 1) ^ 2 * (a (m - 1) ^ 2 + a m ^ 2 + a 1 ^ 2 + a 2 ^ 2) := by ring_nf rw [h₅₁, mul_assoc 2 _ (a (m + 1) ^ 2), mul_comm (∑ x ∈ Finset.range m, a (x + 1) ^ 2), ← mul_assoc 2] have h₅₂: a (m - 1) ^ 2 + a m ^ 2 + a 1 ^ 2 + a 2 ^ 2 ≤ ∑ x ∈ Finset.range m, a (x + 1) ^ 2 := by exact aux1 a m hm₀ refine mul_le_mul ?_ ?_ ?_ ?_ . exact le_of_eq (by rfl) . exact h₅₂ . exact _root_.zero_le (a (m - 1) ^ 2 + a m ^ 2 + a 1 ^ 2 + a 2 ^ 2) . exact _root_.zero_le (2 * a (m + 1) ^ 2) have h₅₃: ∑ x ∈ Finset.range (m - 2), (a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) + a (m - 1) ^ 4 + 2 * a (m - 1) ^ 2 * a m ^ 2 + a m ^ 4 + 2 * a m ^ 2 * a 1 ^ 2 ≤ (∑ x ∈ Finset.range m, a (x + 1) ^ 2) ^ 2 := by have h₅₄: ∑ x ∈ Finset.range (m - 2), (a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) + a (m - 1) ^ 4 + 2 * a (m - 1) ^ 2 * a m ^ 2 + a m ^ 4 + 2 * a m ^ 2 * a 1 ^ 2 ≤ ∑ x ∈ Finset.range (m - 2), (a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) + (a (m - 1) ^ 4 + 2 * a (m - 1) ^ 2 * a m ^ 2 + 2 * a (m - 1) ^ 2 * a 1 ^ 2) + (a m ^ 4 + 2 * a m ^ 2 * a 1 ^ 2 + 2 * a m ^ 2 * a 2 ^ 2) := by repeat rw [add_assoc] repeat refine add_le_add_left ?_ _ have h₅₅: 2 * a (m - 1) ^ 2 * a 1 ^ 2 + (a m ^ 4 + (2 * a m ^ 2 * a 1 ^ 2 + 2 * a m ^ 2 * a 2 ^ 2)) = (a m ^ 4 + 2 * a m ^ 2 * a 1 ^ 2) + (2 * a (m - 1) ^ 2 * a 1 ^ 2 + 2 * a m ^ 2 * a 2 ^ 2) := by ring_nf rw [h₅₅] exact le_self_add exact le_trans h₅₄ hm₁ apply add_le_add h₅₃ at h₅₀ have h₅₆: ∑ x ∈ Finset.range (m - 2), (a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) + a (m - 1) ^ 4 + 2 * a (m - 1) ^ 2 * a m ^ 2 + a m ^ 4 + 2 * a m ^ 2 * a 1 ^ 2 + (2 * a (m - 1) ^ 2 * a (m + 1) ^ 2 + 2 * a m ^ 2 * a (m + 1) ^ 2 + 2 * a (m + 1) ^ 2 * a 1 ^ 2 + 2 * a (m + 1) ^ 2 * a 2 ^ 2 + a (m + 1) ^ 4) = ∑ x ∈ Finset.range (m - 2), (a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) + (a (m - 1) ^ 4 + 2 * a (m - 1) ^ 2 * a m ^ 2 + 2 * a (m - 1) ^ 2 * a (m + 1) ^ 2) + (a m ^ 4 + 2 * a m ^ 2 * a (m + 1) ^ 2 + 2 * a m ^ 2 * a 1 ^ 2) + (a (m + 1) ^ 4 + 2 * a (m + 1) ^ 2 * a 1 ^ 2 + 2 * a (m + 1) ^ 2 * a 2 ^ 2) := by repeat rw [add_assoc] simp ring_nf rw [← h₅₆] exact h₅₀ theorem imo_2007_p6 (a : ℕ → NNReal) (h₀ : ∑ x ∈ Finset.range 100, ((a (x + 1)) ^ 2) = 1) (h₁ : ∀ x y, x % 100 = y % 100 → a (x + 1) = a (y + 1)) : ∑ x ∈ Finset.range (99), ((a (x + 1)) ^ 2 * a (x + 2)) + (a 100) ^ 2 * a 1 < (12:NNReal) / (25:NNReal) := by have h₂: ∀ x, 2 * a x ^ 2 * a (x + 1) * a (x + 2) ≤ (a x * a (x + 1)) ^ 2 + (a x * a (x + 2)) ^ 2 := by intro x have h₂₀: 2 * (a x * a (x + 1)) * (a x * a (x + 2)) ≤ (a x * a (x + 1)) ^ 2 + (a x * a (x + 2)) ^ 2 := by exact two_mul_le_add_sq (a x * a (x + 1)) (a x * a (x + 2)) have h₂₁: 2 * (a x * a (x + 1)) * (a x * a (x + 2)) = 2 * a x ^ 2 * a (x + 1) * a (x + 2) := by rw [pow_two] ring_nf exact le_of_eq_of_le (id (Eq.symm h₂₁)) h₂₀ have h₃: ∀ x ∈ Finset.range 100, a (x + 1) ≤ 1 := by intros x hx₀ by_contra hx₁ push_neg at hx₁ let fsx : Finset ℕ := {x} have hx₂: 1 < ∑ x ∈ range 100, a (x + 1) ^ 2 := by have hx₃: 0 ≤ ∑ x ∈ (range 100 \ fsx), a (x + 1) ^ 2 := by exact _root_.zero_le (∑ x ∈ range 100 \ fsx, a (x + 1) ^ 2) have hx₄: 1 < ∑ x ∈ (fsx), a (x + 1) ^ 2 := by rw [Finset.sum_singleton] refine one_lt_pow₀ hx₁ ?_ norm_num have hx₅: ∑ x ∈ (range 100 \ fsx), a (x + 1) ^ 2 + ∑ x ∈ (fsx), a (x + 1) ^ 2 = ∑ x ∈ range 100, a (x + 1) ^ 2 := by rw [← Finset.sum_union ?_] . rw [Finset.sdiff_union_self_eq_union] have hx₆: range 100 ∪ fsx = range 100 := by refine Finset.union_eq_left.mpr ?_ exact singleton_subset_iff.mpr hx₀ rw [hx₆] . exact sdiff_disjoint rw [← hx₅] exact lt_add_of_nonneg_of_lt hx₃ hx₄ simp_all only [mem_range, lt_self_iff_false] have h₄: (∑ x ∈ Finset.range 100, (a (x + 2) * (a (x + 1) ^ 2 + 2 * a (x + 2) * a (x + 3)))) ^ 2 ≤ ∑ x ∈ Finset.range 100, (a (x + 1) ^ 4 + 6 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) := by have h₄₀: (∑ x ∈ Finset.range 100, (a (x + 2) * (a (x + 1) ^ 2 + 2 * a (x + 2) * a (x + 3)))) ^ 2 ≤ (∑ x ∈ Finset.range 100, (a (x + 2) ^ 2)) * (∑ x ∈ Finset.range 100, ((a (x + 1) ^ 2 + 2 * a (x + 2) * a (x + 3))) ^ 2) := by refine sum_mul_sq_le_sq_mul_sq (range 100) (fun i => a (i + 2)) _ have h₄₁: ∑ x ∈ Finset.range 100, (a (x + 2) ^ 2) = 1 := by rw [Finset.sum_range_succ'] at h₀ simp at h₀ rw [Finset.sum_range_succ] have h₄₁₁: a 1 = a 101 := by exact h₁ 0 100 rfl rw [← h₄₁₁] exact h₀ have h₄₂: ∑ x ∈ Finset.range 100, ((a (x + 1) ^ 2 + 2 * a (x + 2) * a (x + 3))) ^ 2 = ∑ x ∈ Finset.range 100, ((a (x + 1) ^ 4 + 4 * a (x + 1) ^ 2 * a (x + 2) * a (x + 3) + 4 * a (x + 2) ^ 2 * a (x + 3) ^ 2)) := by refine Finset.sum_congr (rfl) ?_ intros x _ rw [add_sq] ring_nf rw [h₄₁, one_mul, h₄₂] at h₄₀ have h₄₃: ∑ x ∈ Finset.range 100, ((a (x + 1) ^ 4 + 4 * a (x + 1) ^ 2 * a (x + 2) * a (x + 3) + 4 * a (x + 2) ^ 2 * a (x + 3) ^ 2)) ≤ ∑ x ∈ Finset.range 100, ((a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * (a (x + 2) ^ 2 + a (x + 3) ^ 2) + 4 * a (x + 2) ^ 2 * a (x + 3) ^ 2)) := by refine Finset.sum_le_sum ?_ intros x _ rw [add_comm (a (x + 1) ^ 4) _, add_comm (a (x + 1) ^ 4) _] rw [add_assoc, add_assoc] refine add_le_add ?_ ?_ . have hx₁: 2 * a (x + 1) ^ 2 * a (x + 1 + 1) * a (x + 1 + 2) ≤ (a (x + 1) * a (x + 1 + 1)) ^ 2 + (a (x + 1) * a (x + 1 + 2)) ^ 2 := by exact h₂ (x + 1) have hx₂: 2 * a (x + 1) ^ 2 * a (x + 2) * a (x + 3) ≤ a (x + 1) ^ 2 * (a (x + 2) ^ 2 + a (x + 3) ^ 2) := by rw [mul_add] refine le_of_le_of_eq hx₁ ?_ ring_nf have hx₃: (4:NNReal) = 2 * 2 := by norm_num rw [hx₃] repeat rw [mul_assoc] have hx₄: 0 < (2:NNReal) := by norm_num refine (mul_le_mul_left hx₄).mpr ?_ ring_nf ring_nf at hx₂ exact hx₂ . exact Preorder.le_refl (a (x + 1) ^ 4 + 4 * a (x + 2) ^ 2 * a (x + 3) ^ 2) have h₄₄: ∑ x ∈ Finset.range 100, ((a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * (a (x + 2) ^ 2 + a (x + 3) ^ 2) + 4 * a (x + 2) ^ 2 * a (x + 3) ^ 2)) = ∑ x ∈ Finset.range 100, (a (x + 1) ^ 4 + 6 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) := by rw [Finset.sum_add_distrib] have h₄₄₁: ∑ x ∈ range 100, 4 * a (x + 2) ^ 2 * a (x + 3) ^ 2 = ∑ x ∈ range 100, 4 * a (x + 1) ^ 2 * a (x + 2) ^ 2 := by rw [Finset.sum_range_succ _ 99, sum_range_succ' _ 99] have g₀: a 101 = a 1 := by exact h₁ 100 0 rfl have g₁: a 102 = a 2 := by exact h₁ 101 1 rfl rw [g₀, g₁] rw [h₄₄₁, ← Finset.sum_add_distrib] refine Finset.sum_congr (rfl) ?_ intros x _ rw [mul_add] ring_nf rw [h₄₄] at h₄₃ exact le_trans h₄₀ h₄₃ have h₆: ∑ x ∈ range 100, 4 * a (x + 1) ^ 2 * a (x + 2) ^ 2 ≤ 1 := by have h₆₀: ∑ x ∈ range 100, 4 * a (x + 1) ^ 2 * a (x + 2) ^ 2 = ∑ x ∈ range 100, 4 * (a (x + 1) ^ 2 * a (x + 2) ^ 2) := by refine Finset.sum_congr rfl ?_ intros x _ ring_nf rw [h₆₀, ← Finset.mul_sum] let fs₂ := Finset.range (100) let fs₀ : Finset ℕ := fs₂.filter (fun x => Odd x) let fs₁ : Finset ℕ := fs₂.filter (fun x => Even x) have h₆₁ : Disjoint fs₀ fs₁ := by refine Finset.sdiff_eq_self_iff_disjoint.mp (by rfl) have h₆₂ : fs₀ ∪ fs₁ = fs₂ := by symm refine Finset.ext_iff.mpr ?_ intro a constructor . intro ha₀ refine mem_union.mpr ?mp.a have ha₁: Odd a ∨ Even a := by exact Or.symm (even_or_odd a) cases' ha₁ with ha₂ ha₃ . left refine mem_filter.mpr ?mp.a.inl.h.a exact And.symm ⟨ha₂, ha₀⟩ . right refine mem_filter.mpr ?mp.a.inl.h.b exact And.symm ⟨ha₃, ha₀⟩ . intro ha₀ apply mem_union.mp at ha₀ cases' ha₀ with ha₁ ha₂ . exact mem_of_mem_filter a ha₁ . exact mem_of_mem_filter a ha₂ have h₆₃: 4 * ∑ i ∈ fs₂, a (i + 1) ^ 2 * a (i + 2) ^ 2 ≤ 4 * ((∑ i ∈ fs₀, (a (i + 1) ^ 2)) * (∑ i ∈ fs₁, (a (i + 1) ^ 2))) := by refine mul_le_mul (by norm_num) ?_ ?_ (by norm_num) . rw [← h₆₂, Finset.sum_union h₆₁] have g₀: ∑ i ∈ fs₁, a (i + 1) ^ 2 = ∑ i ∈ fs₀, (a i) ^ 2 := by refine sum_bij ?_ ?h.b2 ?h.b3 ?h.b4 ?h.b5 . intros b _ exact (b + 1) . intros b hb₀ apply mem_filter.mp at hb₀ cases' hb₀ with hb₀ hb₁ have hb₂: Odd (b + 1) := by exact Even.add_one hb₁ have hb₃: b ≤ 98 := by by_contra hc₀ apply mem_range.mp at hb₀ interval_cases b have hc₁: ¬ Even 99 := by decide exact hc₁ hb₁ have hb₄: b + 1 < 100 := by linarith have hb₅: (b + 1) ∈ fs₂ := by exact mem_range.mpr hb₄ refine mem_filter.mpr ?_ exact And.symm ⟨hb₂, hb₅⟩ . intros b _ c _ hb₂ linarith . intros b hb₀ use (b - 1) refine exists_prop.mpr ?h.a have hb₁: b ∈ fs₂ ∧ Odd b := by exact mem_filter.mp hb₀ have hb₂: 1 ≤ b := by by_contra hc interval_cases b have hb₃: ¬ Odd 0 := by decide exact hb₃ hb₁.2 constructor . cases' hb₁ with hb₁ hb₃ have hb₄: Even (b - 1) := by exact Nat.Odd.sub_odd hb₃ (by decide) have hb₅: (b - 1) ∈ fs₂ := by refine mem_range.mpr ?_ have hb₆: b < 100 := by exact List.mem_range.mp hb₁ omega refine mem_filter.mpr ?_ exact And.symm ⟨hb₄, hb₅⟩ . exact Nat.sub_add_cancel hb₂ . exact fun a_1 _ => rfl have g₁: ∑ x ∈ fs₁, a (x + 1) ^ 2 * a (x + 2) ^ 2 = ∑ x ∈ fs₀, a (x) ^ 2 * a (x + 1) ^ 2 := by refine sum_bij ?_ ?_ ?_ ?_ ?_ . intros b _ exact (b + 1) . intros b hb₀ apply mem_filter.mp at hb₀ cases' hb₀ with hb₀ hb₁ have hb₂: Odd (b + 1) := by exact Even.add_one hb₁ have hb₃: b ≤ 98 := by by_contra hc₀ apply mem_range.mp at hb₀ interval_cases b have hc₁: ¬ Even 99 := by decide exact hc₁ hb₁ have hb₄: b + 1 < 100 := by linarith have hb₅: (b + 1) ∈ fs₂ := by exact mem_range.mpr hb₄ refine mem_filter.mpr ?_ exact And.symm ⟨hb₂, hb₅⟩ . intros b _ c _ hb₂ linarith . intros b hb₀ use (b - 1) refine exists_prop.mpr ?h.b have hb₁: b ∈ fs₂ ∧ Odd b := by exact mem_filter.mp hb₀ have hb₂: 1 ≤ b := by by_contra hc interval_cases b have hb₃: ¬ Odd 0 := by decide exact hb₃ hb₁.2 constructor . cases' hb₁ with hb₁ hb₃ have hb₄: Even (b - 1) := by exact Nat.Odd.sub_odd hb₃ (by decide) have hb₅: (b - 1) ∈ fs₂ := by refine mem_range.mpr ?_ have hb₆: b < 100 := by exact List.mem_range.mp hb₁ omega refine mem_filter.mpr ?_ exact And.symm ⟨hb₄, hb₅⟩ . exact Nat.sub_add_cancel hb₂ . exact fun a_1 _ => rfl rw [g₀, g₁, Finset.sum_mul_sum, add_comm, ← sum_add_distrib] refine sum_le_sum ?_ intros x hx₀ apply mem_filter.mp at hx₀ cases' hx₀ with hx₀ hx₁ apply mem_range.mp at hx₀ by_cases hx₃: x < 99 . clear h₀ h₁ h₂ h₃ h₄ h₆₀ g₀ g₁ let fs₃ : Finset ℕ := {x, (x + 2)} have hx₄: fs₃ ⊆ fs₀ := by intros b hb₀ have hb₁: b = x ∨ b = x + 2 := by have g₀: fs₃ = {x, x + 2} := by rfl simp_all only [mem_insert, mem_singleton] cases' hb₁ with hb₁ hb₁ . rw [hb₁] refine mem_filter.mpr ?_ apply mem_range.mpr at hx₀ exact And.symm ⟨hx₁, hx₀⟩ . rw [hb₁] refine mem_filter.mpr ?_ constructor . have hx₄: x < 98 := by by_contra hc interval_cases x have hx₅: ¬ Odd 98 := by decide apply hx₅ hx₁ refine mem_range.mpr ?_ linarith . refine Odd.add_even hx₁ ?_ decide have hx₅: ∑ j ∈ fs₃, a (x + 1) ^ 2 * a j ^ 2 = a (x + 1) ^ 2 * a x ^ 2 + a (x + 1) ^ 2 * a (x + 2) ^ 2 := by have hx₆: fs₃ = {x, x + 2} := by rfl refine Finset.sum_eq_add_of_mem (x) (x + 2) ?_ ?_ (by norm_num) ?_ . rw [hx₆] exact mem_insert_self x {x + 2} . rw [hx₆] simp . intros c hc₀ hc₁ exfalso rw [hx₆] at hc₀ simp only [mem_insert, mem_singleton] at hc₀ have hc₃: ¬ (c ≠ x ∧ c ≠ x + 2) := by omega exact hc₃ hc₁ rw [← Finset.sum_sdiff hx₄, hx₅] refine le_add_left ?_ refine le_of_eq ?_ rw [mul_comm (a x ^ 2) (a (x + 1) ^ 2)] . interval_cases x norm_num have hx₄: a 101 = a 1 := by exact h₁ 100 0 rfl let fs₃: Finset ℕ := {1, 99} have hx₅: fs₃ ⊆ fs₀ := by refine Finset.subset_iff.mpr ?_ intros b hb₀ have hb₁: b = 1 ∨ b = 99 := by exact List.mem_pair.mp hb₀ cases' hb₁ with hb₂ hb₂ . refine mem_filter.mpr ?_ rw [hb₂] constructor . refine mem_range.mpr (by decide) . decide . rw [hb₂] refine mem_filter.mpr ?_ constructor . exact self_mem_range_succ 99 . decide have hx₆: ∑ x ∈ fs₃, a 100 ^ 2 * a x ^ 2 = a 100 ^ 2 * a 99 ^ 2 + a 100 ^ 2 * a 1 ^ 2 := by clear h₀ h₁ h₂ h₃ h₄ h₆₀ have hx₇: fs₃ = {1, 99} := by rfl refine Finset.sum_eq_add_of_mem (99:ℕ) (1:ℕ) ?_ ?_ (by norm_num) ?_ . rw [hx₇] decide . rw [hx₇] decide . intros c hc₀ hc₁ exfalso have hc₂: c = 99 ∨ c = 1 := by refine Or.symm ?_ exact List.mem_pair.mp hc₀ have hc₃: ¬ (c ≠ 99 ∧ c ≠ 1) := by omega exact hc₃ hc₁ rw [← Finset.sum_sdiff hx₅, hx₄, hx₆] refine le_add_left ?_ refine le_of_eq ?_ rw [mul_comm (a 99 ^ 2) (a 100 ^ 2)] . exact _root_.zero_le (∑ i ∈ range 100, a (i + 1) ^ 2 * a (i + 2) ^ 2) have h₆₄: 4 * ((∑ i ∈ fs₀, (a (i + 1) ^ 2)) * (∑ i ∈ fs₁, (a (i + 1) ^ 2))) ≤ (∑ i ∈ fs₀, (a (i + 1) ^ 2) + ∑ i ∈ fs₁, (a (i + 1) ^ 2)) ^ 2 := by have g₀: ∀ x y : ℝ, 4 * x * y ≤ (x + y) ^ 2 := by intros x y rw [add_sq] have g₁: 2 * x * y ≤ x ^ 2 + y ^ 2 := by exact two_mul_le_add_sq x y linarith rw [← mul_assoc] let x := (∑ i ∈ fs₀, a (i + 1) ^ 2) let y := (∑ i ∈ fs₁, a (i + 1) ^ 2) refine g₀ x y have h₆₅: (∑ i ∈ fs₀, (a (i + 1) ^ 2) + ∑ i ∈ fs₁, (a (i + 1) ^ 2)) ^ 2 = 1 := by rw [← Finset.sum_union h₆₁, h₆₂, h₀] exact one_pow 2 refine le_trans h₆₃ ?_ refine le_trans h₆₄ ?_ rw [h₆₅] let S : NNReal := ∑ x ∈ Finset.range 99, ((a (x + 1)) ^ 2 * a (x + 2)) + (a 100) ^ 2 * a 1 have hS : S = ∑ x ∈ Finset.range 99, ((a (x + 1)) ^ 2 * a (x + 2)) + (a 100) ^ 2 * a 1 := by rfl rw [← hS] have hS₁ : S = ∑ x ∈ Finset.range 100, ((a (x + 1)) ^ 2 * a (x + 2)) := by rw [Finset.sum_range_succ] norm_num have g₀: a 101 = a 1 := by exact h₁ 100 0 rfl rw [g₀] have h₇: (3 * S) ^ 2 ≤ 2 := by have h₇₀: 3 * S = ∑ x ∈ Finset.range 100, (a (x + 2) * (a (x + 1) ^ 2 + 2 * a (x + 2) * a (x + 3))) := by have g₀: ∑ x ∈ Finset.range 100, (a (x + 2) * (a (x + 1) ^ 2 + 2 * a (x + 2) * a (x + 3))) = ∑ x ∈ Finset.range 100, (a (x + 1) ^ 2 * a (x + 2) + 2 * a (x + 2) ^ 2 * a (x + 3)) := by refine Finset.sum_congr rfl ?_ intros x _ ring_nf have g₁: (3:NNReal) = 1 + 2 := by norm_num rw [g₀, Finset.sum_add_distrib] rw [g₁, hS₁, add_mul, one_mul, Finset.mul_sum] simp rw [Finset.sum_range_succ' _ 99, sum_range_succ _ 99] norm_num have g₂: a 101 = a 1 := by exact h₁ 100 0 rfl have g₃: a 102 = a 2 := by exact h₁ 101 1 rfl rw [g₂, g₃, ← mul_assoc 2] simp refine Finset.sum_congr rfl ?_ intros x _ ring_nf rw [← h₇₀] at h₄ refine le_trans h₄ ?_ have h₇₁: ∑ x ∈ range 100, (a (x + 1) ^ 4 + 6 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) = ∑ x ∈ range 100, (a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) + ∑ x ∈ range 100, 4 * a (x + 1) ^ 2 * a (x + 2) ^ 2 := by rw [← Finset.sum_add_distrib] refine Finset.sum_congr rfl ?_ intros x _ ring_nf have h₇₂: ∑ x ∈ range 100, (a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) ≤ 1 := by refine le_trans (aux2 a 100 ?_ h₁) ?_ . omega . refine (sq_le_one_iff₀ ?_).mpr ?_ . exact _root_.zero_le (∑ x ∈ range 100, a (x + 1) ^ 2) . rw [← h₀] rw [h₇₁, ← one_add_one_eq_two] refine add_le_add ?_ h₆ norm_num exact h₇₂ have h₈ : S ≤ (NNReal.sqrt 2) / (3:NNReal) := by have h₆₀: NNReal.sqrt (((3:NNReal) * S) ^ 2) ≤ NNReal.sqrt 2 := by exact NNReal.sqrt_le_sqrt.mpr h₇ rw [sqrt_sq, mul_comm] at h₆₀ refine (le_div_iff₀ (by norm_num)).mpr h₆₀ have h₉: (NNReal.sqrt 2) / (3:NNReal) < (12:NNReal) / (25:NNReal) := by have h₇₁: 2 < 144 / (625:NNReal) * 9 := by refine (one_lt_div (by norm_num)).mp ?_ rw [mul_comm_div, ← mul_div_assoc, div_div] norm_num refine (one_lt_div (by norm_num)).mpr ?_ norm_num have h₇₂: (NNReal.sqrt 2 / 3:NNReal) ^ 2 < (12 / 25:NNReal) ^ 2 := by rw [div_pow, div_pow] norm_num refine (div_lt_iff₀ ?_).mpr h₇₁ exact ofNat_pos' have h₇₃: NNReal.sqrt ((NNReal.sqrt 2 / 3) ^ 2) < NNReal.sqrt ((12 / 25) ^ 2) := by exact sqrt_lt_sqrt.mpr h₇₂ rw [sqrt_sq, sqrt_sq] at h₇₃ exact h₇₃ exact lt_of_le_of_lt h₈ h₉