Question 25
(USA 1) Suppose that $n\ge2$ and $x_1,x_2,\ldots,x_n$ are real numbers between 0 and 1 (inclusive). Prove that for some index $i$ between 1 and $n-1$ the inequality
$$x_i(1-x_{i+1})\ge\frac14x_1(1-x_n)$$
holds.
$$x_i(1-x_{i+1})\ge\frac14x_1(1-x_n)$$
holds.