5. (3 points) Petya and Vasya are playing a game. There are a total of three moves in the game. On the first move, Petya breaks a stick that is 10 cm long into two parts. Then Vasya breaks one of the resulting sticks into two parts. On the last move, Petya breaks one of the three resulting sticks into two parts. Vasya wins if a triangle can be formed from any three of the resulting pieces, and Petya wins otherwise. Who will win with correct play?
Answer: Petya.

Solution: Petya's first move is to break the original stick into two 5 cm sticks. Vasya then breaks one of these sticks into two parts, with lengths $a$ and $b$. Assume $a \geqslant b$. Petya then breaks the stick of length $b$ into two arbitrary parts $c$ and $d$.

The 5 cm segment is equal to the sum of the three remaining segments, so it cannot form a triangle with any pair of the others. The segment $a$ is not less than the sum of $c+d$, so a triangle cannot be formed from $a, c$, and $d$ either.