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Partition into sets with equal sum of powers in F_p

Source: IMOC 2021 N7

August 11, 2021
modular arithmeticnumber theory

Problem Statement

Let pp be a given odd prime. Find the largest integer k′k' such that it is possible to partition {1,2,⋯ ,p−1}\{1,2,\cdots,p-1\} into two sets X,YX,Y such that for any kk with 0≤k≤k′0 \le k \le k',
∑a∈Xak≡∑b∈Ybk(modp)\sum_{a \in X}a^k \equiv \sum_{b \in Y}b^k \pmod p
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