Problem #PRU-105052

Problem

In a chess tournament, each participant played two games with each of the other participants: one with white pieces, the other with black. At the end of the tournament, it turned out that all of the participants scored the same number of points (1 point for a victory, $\frac{1}{2}$ a point for a draw and 0 points for a loss). Prove that there are two participants who have won the same number of games using white pieces.