Number theory · Parity · Permutations · Congruences
Let \(a_{1}, a_{2}, \ldots, a_{n}\) be pairwise distinct numbers from the set \(\{1, 2, \ldots, n\}\), where \(n \in \mathbb{N}\). Prove that the number
\[ (a_{1} - 1) + (a_{2} - 2)^{2} + \cdots + (a_{n} - n)^{n} \]
is always even.
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Serbian Municipal Competition 2026, high school grade I, category A, problem 2. Organized by the Mathematical Society of Serbia (DMS). Source