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Algebra Difficulty 6.4 National olympiad Prove it

25. (USA 1) Suppose that n≥2n \geq 2 and x1,x2,…,xnx_{1}, x_{2}, \ldots, x_{n} are real numbers between 0 and 1 (inclusive). Prove that for some index ii between 1 and n−1n-1 the inequality
xi(1−xi+1)≥14x1(1−xn) x_{i}\left(1-x_{i+1}\right) \geq \frac{1}{4} x_{1}\left(1-x_{n}\right)
holds.

Solution

25. Since replacing x1x_{1} by 1 can only reduce the set of indices ii for which the desired inequality holds, we may assume x1=1x_{1}=1. Similarly we may assume xn=0x_{n}=0. Now we can let ii be the largest index such that xi>1/2x_{i}>1 / 2. Then xi+1≤1/2x_{i+1} \leq 1 / 2, hence xi(1−xi+1)≥14=14x1(1−xn). x_{i}\left(1-x_{i+1}\right) \geq \frac{1}{4}=\frac{1}{4} x_{1}\left(1-x_{n}\right) .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.