Question 44
Let $p \ge 3$ be a prime number. There are $n = 2p$ integers on the blackboard. A student chooses one or more numbers and writes down their sum modulo $n$. Suppose that the remainders $1, 2, \dots, p-1, p+1, \dots, n-1$ appear the same number of times when the student writes down all $2^n - 1$ possible sums modulo $n$. Prove that the sum of the numbers on the blackboard is divisible by $p$.
(Proposed by Bayarmagnai Gombodorj)
(Proposed by Bayarmagnai Gombodorj)