Question 184
Let $n \ge 2$ be a positive integer. Petya and Vasya play the following game. Petya chooses $2n$ (not necessarily distinct) nonnegative real numbers $x_1, x_2, \dots, x_{2n}$ whose sum equals $1$, and tells those numbers to Vasya. Vasya arranges those numbers in a circle by his own choice, computes the product in each pair of adjacent numbers, and writes down the maximal of those products. Petya wants to maximize the written number, while Vasya wants to minimize it. What number will be written at an optimal game? (A. Khrabrov)