Problem=

Let $n_1, n_2, \dots , n_m$ be integers where $m$ is odd. Let $x = (x_1, \dots , x_m)$ denote a permutation of the integers $1, 2, \cdots , m$ . Let $f(x) = x_1n_1 + x_2n_2 + ... + x_mn_m$ . Show that for some distinct permutations $a$ , $b$ the difference $f(a) - f(b)$ is a multiple of $m!$ .

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2001 IMO Problems/Problem 4

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