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x_i+1 <= 0.5

Source: IMO ShortList 1991, Problem 25 (USA 1)

August 15, 2008
SequenceInequalityinequality systemalgebraIMO Shortlist

Problem Statement

Suppose that n≥2 n \geq 2 and x1,x2,…,xn x_1, x_2, \ldots, x_n are real numbers between 0 and 1 (inclusive). Prove that for some index i i between 1 1 and n \minus{} 1 the inequality x_i (1 \minus{} x_{i\plus{}1}) \geq \frac{1}{4} x_1 (1 \minus{} x_{n})
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