Compfiles: Catalog Of Math Problems Formalized In Lean

Imo1966P4


import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Tactic

/-!
# International Mathematical Olympiad 1966, Problem 4

Prove that for every natural number n and for every real
number x that is not of the form kπ/2ᵗ for t a non-negative
integer and k any integer,

 1 / (sin 2x) + 1 / (sin 4x) + ... + 1 / (sin 2ⁿx) = cot x - cot 2ⁿ x.
-/

namespace Imo1966P4

open BigOperators

theorem imo1966_p4 (n : ℕ) (x : ℝ)
    (hx : ∀ t : ℕ, ∀ k : ℤ, x ≠ k * Real.pi / 2^t) :
    ∑ i ∈ Finset.range n, 1 / Real.sin (2^(i + 1) * x) =
    1 / Real.tan x - 1 / Real.tan (2^n * x) := sorry

This problem does not yet have a complete formalized solution.

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