Problem

Source: INAMO 2019 P8

Tags: combinatorics



Let $n > 1$ be a positive integer and $a_1, a_2, \dots, a_{2n} \in \{ -n, -n + 1, \dots, n - 1, n \}$. Suppose \[ a_1 + a_2 + a_3 + \dots + a_{2n} = n + 1 \]Prove that some of $a_1, a_2, \dots, a_{2n}$ have sum 0.