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1c3ffd8 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 | import Mathlib
set_option linter.unusedVariables.analyzeTactics true
open Real BigOperators
theorem imo_1969_p2
(m n : ℝ)
(k : ℕ)
(a : ℕ → ℝ)
(f : ℝ → ℝ)
-- (h₀ : 0 < k)
-- (h₁ : ∀ x, f x = ∑ i in Finset.range k, ((Real.cos (a i + x)) / (2^i)))
(h₁ : ∀ x, f x = Finset.sum (Finset.range k) fun i => ((Real.cos (a i + x)) / (2^i)))
(h₂ : f m = 0)
(h₃ : f n = 0)
(h₄: Finset.sum (Finset.range k) (fun i => (((cos (a i)) / (2 ^ i)))) ≠ 0) :
∃ t : ℤ, m - n = t * π := by
let Ccos := Finset.sum (Finset.range k) (fun i => (((cos (a i)) / (2 ^ i))))
let Csin := Finset.sum (Finset.range k) (fun i => (((sin (a i)) / (2 ^ i))))
have hCcos: Ccos = Finset.sum (Finset.range k) (fun i => (((cos (a i)) / (2 ^ i)))) := by
exact rfl
have hCsin: Csin = Finset.sum (Finset.range k) (fun i => (((sin (a i)) / (2 ^ i)))) := by
exact rfl
have h₅: ∀ x, f x = Ccos * cos x - Csin * sin x := by
intro x
rw [h₁ x]
have h₅₀: ∑ i ∈ Finset.range k, (cos (a i + x) / 2 ^ i)
= ∑ i ∈ Finset.range k, (((cos (a i) * cos (x) - sin (a i) * sin (x)) / (2^i))) := by
refine Finset.sum_congr (by rfl) ?_
simp
intros i _
refine (div_eq_div_iff ?_ ?_).mpr ?_
. exact Ne.symm (NeZero.ne' (2 ^ i))
. exact Ne.symm (NeZero.ne' (2 ^ i))
. refine mul_eq_mul_right_iff.mpr ?_
simp
exact cos_add (a i) x
rw [h₅₀]
ring_nf
rw [Finset.sum_sub_distrib]
have h₅₂: ∑ i ∈ Finset.range k, cos (a i) * cos x * (1 / 2) ^ i
= ∑ i ∈ Finset.range k, (cos (a i) * (1 / 2) ^ i) * cos x := by
refine Finset.sum_congr (by rfl) ?_
simp
intro i _
linarith
have h₅₃: ∑ x_1 ∈ Finset.range k, sin (a x_1) * sin x * (1 / 2) ^ x_1
= ∑ x_1 ∈ Finset.range k, ((sin (a x_1) * (1 / 2) ^ x_1) * sin x) := by
refine Finset.sum_congr (by rfl) ?_
simp
intro i _
linarith
rw [h₅₂, ← Finset.sum_mul _ _ (cos x)]
rw [h₅₃, ← Finset.sum_mul _ _ (sin x)]
ring_nf at hCcos
ring_nf at hCsin
rw [hCcos, hCsin]
have h₆: (∃ x, (f x = 0 ∧ cos x = 0)) → ∀ y, f y = Ccos * cos y := by
intro g₀
obtain ⟨x, hx₀, hx₁⟩ := g₀
have g₁: Finset.sum (Finset.range k) (fun i => (((sin (a i)) / (2 ^ i)))) = 0 := by
rw [h₅ x, hx₁] at hx₀
simp at hx₀
cases' hx₀ with hx₂ hx₃
. exact hx₂
. exfalso
apply sin_eq_zero_iff_cos_eq.mp at hx₃
cases' hx₃ with hx₃ hx₄
. linarith
. linarith
intro y
rw [h₅ y]
have g₂: Csin = 0 := by
linarith
rw [g₂, zero_mul]
exact sub_zero (Ccos * cos y)
by_cases hmn: (cos m = 0) ∨ (cos n = 0)
. have h₇: ∀ (x : ℝ), f x = Ccos * cos x := by
refine h₆ ?_
cases' hmn with hm hn
. use m
. use n
have h₈: ∀ x, f x = 0 → cos x = 0 := by
intros x hx₀
rw [h₇ x] at hx₀
refine eq_zero_of_ne_zero_of_mul_left_eq_zero ?_ hx₀
exact h₄
have hm₀: ∃ t:ℤ , m = (2 * ↑ t + 1) * π / 2 := by
refine cos_eq_zero_iff.mp ?_
exact h₈ m h₂
have hn₀: ∃ t:ℤ , n = (2 * ↑ t + 1) * π / 2 := by
refine cos_eq_zero_iff.mp ?_
exact h₈ n h₃
obtain ⟨tm, hm₁⟩ := hm₀
obtain ⟨tn, hn₁⟩ := hn₀
rw [hm₁, hn₁]
use (tm - tn)
rw [Int.cast_sub]
ring_nf
. push_neg at hmn
have h₇: tan m = tan n := by
have h₇₀: ∀ (x:ℝ), (f x = 0 ∧ cos x ≠ 0) → tan x = Ccos / Csin := by
intro x hx₀
rw [tan_eq_sin_div_cos]
symm
refine (div_eq_div_iff ?_ ?_).mp ?_
. simp
exact hx₀.2
. simp
have hx₁: Ccos * cos x ≠ 0 := by
refine mul_ne_zero ?_ hx₀.2
exact h₄
have hx₂: Ccos * cos x = Csin * sin x := by
rw [h₅ x] at hx₀
refine eq_of_sub_eq_zero ?_
exact hx₀.1
have hx₃: Csin * sin x ≠ 0 := by
rw [← hx₂]
exact hx₁
exact left_ne_zero_of_mul hx₃
. simp
symm
refine eq_of_sub_eq_zero ?_
rw [h₅ x] at hx₀
linarith
have h₇₁: tan m = Ccos / Csin := by
refine h₇₀ m ?_
constructor
. exact h₂
. exact hmn.1
have h₇₂: tan n = Ccos / Csin := by
refine h₇₀ n ?_
constructor
. exact h₃
. exact hmn.2
rw [h₇₁, h₇₂]
have h₈: sin (m - n) = 0 := by
have h₈₀: tan m - tan n = 0 := by exact sub_eq_zero_of_eq h₇
have h₈₁: (sin m * cos n - cos m * sin n) / (cos m * cos n) = 0 := by
rw [← div_sub_div (sin m) (sin n) hmn.1 hmn.2]
repeat rw [← tan_eq_sin_div_cos]
exact h₈₀
have h₈₂: sin (m - n) / (cos m * cos n) = 0 := by
rw [sin_sub]
exact h₈₁
apply div_eq_zero_iff.mp at h₈₂
cases' h₈₂ with h₈₂ h₈₃
. exact h₈₂
. exfalso
simp at h₈₃
cases' h₈₃ with h₈₄ h₈₅
. exact hmn.1 h₈₄
. exact hmn.2 h₈₅
apply sin_eq_zero_iff.mp at h₈
let ⟨t, ht⟩ := h₈
use t
exact ht.symm
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