Nemotron-IMO-Bench 2026 C11

Let $n$ be a positive integer. A thief has $2n$ accomplices. Several policemen try to catch him, by placing his accomplices under surveillance. In the beginning no accomplice is followed by the policemen. Every morning each policeman places under his surveillance one of the accomplices and every evening the thief stops trusting one of his accomplices. Prove that if less than $2^n$ policemen are involved, then the thief can always make sure that by the $n$th evening no policeman shadows exactly those $n$ accomplices who are still trusted by the thief.