There are $n\geq 12$ points on a circle, each colored black or white. When we consider the $5$ neighbors to the left of any point and $5$ to the right, then among the $10$ points, exactly $5$ are black and $5$ white. Prove that $n$ is divisible by $4$.
Suppose there are $k$ black points. Each black point occurs as a neighbor within distance $5$ for $10$ other points. On the other hand, each point has $5$ black neighbors within distance $5$. So $10k=5n$, which implies $k=n/2$. In particular, $n$ is divisible by $2$.
How can we obtain $n$ divisible by $4$?
We just have to prove there is an even number of black balls. Consider the graph where the vertices are the balls and two balls are joined by an edge if they are at distance $5$ or less and at least one of them is white. The white balls have degree $10$, while the black balls have degree $5$, there must be an even number of black balls since the sum of degrees of a graph is even.