10. There are 2000 nodes, and each pair of nodes is connected by a wire. Now, let Varia and Peter take turns to cut these wires, with Varia starting first. She can only cut one wire each time, while Peter can cut 2 or 3 wires. The one who cuts the last wire loses. Who will win in the end?
(1999 Russian Olympiad Problem)
Problem 556
Official solution
10. Since there are wires, and 1999000 is a multiple of 4, Varia cuts 1 wire, and Peter cuts 3 wires, until 4 wires are left and it's Varia's turn to cut. Varia cuts 1, Peter cuts 2, and the last one is cut by Varia, so Peter can definitely win.