5. A stick 10 cm long lies on the table. Petya breaks it into two pieces and puts both resulting sticks on the table. Vasya performs the same operation with one of the sticks lying on the table, then Petya does the same, and so on, taking turns. Petya wants all the resulting sticks to be shorter than 1 cm after 18 breaks. Vasya wants to prevent Petya. Who of them can achieve their goal regardless of the opponent's actions? (I. Rubanov, S. Berlov)
Answer: Vasya. Solution: Mark 9 points dividing the stick into segments of 1 cm. Vasya needs to play in such a way that all these points are broken. Since he has 9 moves, he can do this. In the end, after 18 breaks, 10 sticks of 1 cm in length will be obtained, some of which are broken into smaller pieces. Since the number of breaks not falling on the marked points is only 9, at least one of the centimeter sticks will remain intact, and Petya will not achieve his goal.