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1^{k_1}+2^{k_2}+3^{k_3}+...+(p-1)^{k_{p-1}} is divisible by $p

Source: 2021 Bosnia Herzegovina MO p2, posted as TST inside contest collections, 13.6.2021

October 7, 2022
number theorydivisible

Problem Statement

Let p>2p > 2 be a prime number. Prove that there is a permutation k1,k2,...,kp−1k_1, k_2, ..., k_{p-1} of numbers 1,2,...,p−11,2,...,p-1 such that the number 1k1+2k2+3k3+...+(p−1)kp−11^{k_1}+2^{k_2}+3^{k_3}+...+(p-1)^{k_{p-1}} is divisible by pp.
Note: The numbers k1,k2,...,kp−1k_1, k_2, ..., k_{p-1} are a permutation of the numbers 1,2,...,p−11,2,...,p-1 if each of of numbers 1,2,...,p−11,2,...,p-1 appears exactly once among the numbers k1,k2,...,kp−1k_1, k_2, ..., k_{p-1}.
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