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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 | import Mathlib
open Real
set_option linter.unusedVariables.analyzeTactics true
lemma sin_mul_cos
(x y : ℝ) :
Real.sin x * Real.cos y = (sin (x + y) + sin (x - y)) / 2 := by
rw [sin_add, sin_sub]
simp
theorem imo_1963_p5 :
Real.cos (π / 7) - Real.cos (2 * π / 7) + Real.cos (3 * π / 7) = 1 / 2 := by
let S:ℝ := Real.cos (π / 7) - Real.cos (2 * π / 7) + Real.cos (3 * π / 7)
have h₀: Real.sin (π / 7) * (S * 2) = Real.sin (π / 7) := by
ring_nf
have h₀₀: sin (π * (1 / 7)) * cos (π * (1 / 7)) * 2 = sin (2 * (π * (1 / 7))) := by
rw [Real.sin_two_mul]
exact (mul_rotate 2 (sin (π * (1 / 7))) (cos (π * (1 / 7)))).symm
rw [h₀₀, sin_mul_cos, sin_mul_cos]
rw [← mul_add, ← mul_sub, ← mul_add, ← mul_sub]
norm_num
ring_nf
have h₀₁: -sin (π * (3 / 7)) + sin (π * (4 / 7)) = 0 := by
rw [add_comm]
refine add_neg_eq_of_eq_add ?_
simp
refine sin_eq_sin_iff.mpr ?_
use 0
right
ring
linarith
have h₁: S = 1 / 2 := by
refine eq_div_of_mul_eq (by norm_num) ?_
nth_rewrite 2 [← mul_one (sin (π / 7))] at h₀
refine (mul_right_inj' ?_).mp h₀
refine sin_ne_zero_iff.mpr ?_
intro n
ring_nf
rw [mul_comm]
simp
push_neg
constructor
. by_contra! hc₀
have hc₁: 7 * (↑n:ℝ) = 1 := by
rw [mul_comm]
exact (mul_eq_one_iff_eq_inv₀ (by norm_num)).mpr hc₀
norm_cast at hc₁
have g₀: 0 < n := by linarith
linarith
. exact pi_ne_zero
exact h₁
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