10. Petya chooses non-negative numbers $x_{1}, x_{2}, \ldots, x_{11}$ such that their sum is 1. Vasya arranges them in a row as he sees fit, calculates the products of adjacent numbers, and writes the largest of the resulting ten products on the board. Petya wants the number on the board to be as large as possible, while Vasya wants it to be as small as possible. What number will end up on the board with the best play by Petya and Vasya? (A. Khryabrov)