67th Romanian Mathematical Olympiad

Denote $a$ the number of the vertices of the upper base which have the first color and $b=n-a$ the number of the vertices of the upper base which have the second color. Then the lower base has $b$ points with the first color and $a$ points with the second color. The number of segments with endpoints differently colored and on different bases is $a^2+b^2-n$. The number of segments with endpoints differently colored and on the same base is $2ab$. So, the total number of segments with endpoints differently colored is $(a+b{)}^2-n=n^2-n$, which is exactly half of the number of all the segments.