5. Each side of the triangle is divided into 2008 equal parts. Through each division point, lines parallel to the other two sides are drawn, resulting in the triangle being divided into equal triangular fields. A row will be called a series of fields enclosed between two adjacent parallel lines, or the single field standing at the vertex of the triangle. Pete and Vasya take turns placing either 1 or -1 in one of the free fields. After all the cells are occupied, the product is calculated in each row. Pete wins if the number of negative products is even, otherwise Vasya wins. Who will win with correct play, if Pete moves first?
Answer Vasya will win. Solution Notice that the triangle has an even number of cells, which means Vasya makes the last move. With his last move, he can always ensure that the product of all the numbers written is equal to -1. We will show that this is enough for his victory. Multiply all the numbers in the rows parallel to one of the sides of the triangle. The resulting product is equal to the product of all the numbers in the triangle and is equal to -1, therefore, the number of rows with negative products among those considered is odd. Considering similarly the rows parallel to the other two sides, we get that the total number of rows with negative products is odd. Thus, Vasya wins.