Question 13
Let $n \ge 2$. Positive integers $a_1, a_2, \dots, a_n$ whose sum is even and which satisfy $a_i \le i$ for every $i = 1, 2, \dots, n$, are given. Prove that it is possible to choose signs in the expression $a_1 \pm a_2 \pm \dots \pm a_n$ in such a way that its value becomes $0$. (Seniors.)