5. There are 2000 nodes, each connected to every other node with a wire. Vasia and Petia take turns cutting these wires. Vasia starts first, and he can only cut one wire at a time, while Petia can cut two or win.

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5. There are 2000 nodes, each connected to every other node with a wire. Vasia and Petia take turns cutting these wires. Vasia starts first, and he can only cut one wire at a time, while Petia can cut two or win.
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Petia can win.
Because there are $C_{2000}^{2}=1999000$ wires, and since 1999000 is a multiple of 4, Vasia cuts one, and Petia cuts three, until four wires are left, at which point it is Vasia's turn to cut. Vasia cuts one, Petia cuts the next two, and the last one is cut by Vasia, so Petia can win.
(Tianjin Normal University Mathematics Department, Jianquan Gong)