Belorusija 2012

Answer: any integer number from $0$ to $N-2$ of the same parity as $N$.

Consider arbitrary arrangement of the boys along the circle. We say that a boy in the given arrangement is tall if he is taller than both of his neighbors, and a boy is short if he is shorter than both of his neighbors.

The numbers of tall and short boys are equal in any arrangement (see solution of Problem C.8). Let $b$ be the number of the tall boys in the arrangement. Then the number $s$ of the middle boys is equal to $N-2b$ and has the same parity with $N$. Since the number $b$ of the tall boys in the arrangement can admit any value from $1\le b\le [N/2]$ (see solution of Problem C.8), the number $s$ of the middle boys can admit any value from $[0;N-2]$ and must be the same parity as $N$. The corresponding examples for any $s$ see in solution of Problem B.8.