Question 160
Petya and Vasya play the following game. Petya chooses 100 (not necessarily distinct) nonnegative real numbers $x_1, x_2, \dots, x_{100}$ whose sum equals 1, and tells those numbers to Vasya. Vasya splits the numbers into 50 pairs by his own choice, computes the product of numbers in each pair, 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?