32nd IMO shortlist 1991/25 solution

Problem 25

Given n > 1 real numbers 0 ≤ x_i ≤ 1, show that for some i < n we have x_i(1 - x_i+1) ≥ x₁(1 - x_n)/4.

Solution

Suppose first that x_n-1 ≥ ½. Then we can take i = n-1, because x_i(1 - x_i+1) ≥ ½ (1 - x_n) > ¼ x₁(1 - x_n). So we may assume x_n-1 < ½.

Now if x_i > ½ for some i < n-1, then we can take the largest i for which this holds. So x_i+1 ≤ ½. Hence x_i(1 - x_i+1) > ½ ½ = ¼ ≥ ¼ x₁(1 - x_n).

If there is no such i, then we must have x₂ ≤ ½. Take i = 1. Then x_i(1 - x_i+1) = x₁(1 - x₂) ≥ ½ x₁ > ¼ x₁(1 - x_n).