OSN 2019 Problem 8

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.