XIV OBM

Let $m$ and $w$ be the number of men and women in the tournament. Each game assigns a total of 1 point to the players, the total of points assigned to games between two men and two women are $\left(\genfrac{}{}{0ex}{}{m}{2}\right)$ and $\left(\genfrac{}{}{0ex}{}{w}{2}\right)$. Since each player scored the same total of points against men as against women and the total of games played is $\left(\genfrac{}{}{0ex}{}{m+w}{2}\right)$, $$2\left(\left(\genfrac{}{}{0ex}{}{m}{2}\right)+\left(\genfrac{}{}{0ex}{}{w}{2}\right)\right)=\left(\genfrac{}{}{0ex}{}{m+w}{2}\right)⟺m+w=(m-w{)}^2,$$ that is, the total number of players, $m+w$, is a perfect square.