IMO 1991 SL 25

Origin: USA

Problem

Suppose that n \geq2 and x1, x2, . . . , xn are real numbers between 0 and 1 (inclusive). Prove that for some index i between 1 and n −1 the inequality xi(1 −xi+1) \geq1 4x1(1 −xn) holds.

Solution

Since replacing x1 by 1 can only reduce the set of indices i for which the desired inequality holds, we may assume x1 = 1. Similarly we may assume xn = 0. Now we can let i be the largest index such that xi > 1/2. Then xi+1 \leq1/2, hence xi(1 −xi+1) \geq1 4 = 1 4x1(1 −xn).