import Mathlib
/-!
# International Mathematical Olympiad 1988, Problem 4
Show that the set of real numbers $x$ which satisfy the inequality
$$\sum_{k=1}^{70} \frac{k}{x - k} \ge \frac{5}{4}$$
is a union of disjoint intervals, the sum of whose lengths is $1988$.
-/
namespace Imo1988P4
open scoped BigOperators
theorem imo1988_p4 :
∃ (n : ℕ) (I : Fin n → Set ℝ),
(∀ i, (I i).OrdConnected) ∧
(Pairwise fun i j ↦ Disjoint (I i) (I j)) ∧
(⋃ i, I i) =
{x : ℝ | (5 : ℝ) / 4 ≤ ∑ k ∈ Finset.Icc (1 : ℕ) 70, (k : ℝ) / (x - k)} ∧
∑ i, MeasureTheory.volume (I i) = 1988 := sorry
end Imo1988P4
This problem does not yet have a complete formalized solution.