Question 17
Let integers $n\ge k\ge3$ be given. There are $n+1$ points in the plane, no three collinear. Color every segment joining two of them with one of $k$ colors. An angle is bicolored when its vertex is one of the points and its two sides are segments of different colors. Prove that some coloring has strictly more than
$$n\left\lfloor n/k\right\rfloor^2\binom{k}{2}$$
bicolored angles. Here $\lfloor t\rfloor$ denotes the greatest integer not exceeding $t$.
$$n\left\lfloor n/k\right\rfloor^2\binom{k}{2}$$
bicolored angles. Here $\lfloor t\rfloor$ denotes the greatest integer not exceeding $t$.