Bakayev E.V.

Petya places 500 kings on the cells of a $100 \times 50$ board so that they do not attack each other. And Vasya places 500 kings on the white cells (in a chessboard coloring) of a $100 \times 100$ board so that they do not attack each other. Who has more ways to do this?
For each of Petya's arrangements, we will correspond it to some arrangement of Vasya, and different arrangements will correspond to different ones. For each of the two methods, there will be a Vasya's arrangement that does not result from this method. For example, when kings occupy all the white cells in some ten rows, pairwise non-adjacent. Therefore, Vasya's arrangements will be more.

The first method. Let's call the right side of Petya's board the axis. Consider any arrangement of Petya's. By reflecting all the kings on the black cells relative to the axis, we get some arrangement of Vasya's: 500 kings on white cells of the $100 \times 100$ board. Indeed, the kings on one half of the board will not attack each other because they did not attack before. And kings from different halves can attack each other only if they are adjacent to the axis of reflection. But the corresponding "black" kings have moved one cell across the axis, so they do not attack the kings from their former column.

The second method (Rashid Aliyev, 10th grade, Makhachkala). We will divide each row of Vasya's board into dominoes. The dominoes form a $100 \times 50$ board. No matter how Petya places 500 kings in the centers of the dominoes, Vasya can shift each king to the white cell of the domino. If after this some two kings attack each other, then they will be in adjacent dominoes, so they attacked each other in Petya's arrangement as well.

## Answer

Vasya.