US-EGMO-TST 2026 Problem 1

Let $n$ be a positive integer. Prove that one can paint the non-zero coefficients of the polynomial

$$f(x_1, x_2, \ldots, x_n) = \prod_{k=0}^{n} (x_1 + x_2 + \cdots + x_n - k)$$

with $2^n - 1$ colors such that the coefficients of each color have sum 0, and each color is used at least once.