Question 12
Let $ n \geq 1 $ be a given integer, and let $ a_1, \ldots, a_n $ be a sequence of integers such that $ n $ divides the sum $ a_1 + \cdots + a_n $. Show that there exist permutations $ \sigma $ and $ \tau $ of 1, 2, $ \ldots $, $ n $ such that $ \sigma(i) + \tau(i) \equiv a_i \pmod{n} $ for all $ i = 1, \ldots, n $.