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Number Theory

Source: Iranian TST 2020, second exam day 2, problem 6

March 12, 2020
number theoryprime numbers

Problem Statement

pp is an odd prime number. Find all p−12\frac{p-1}2-tuples (x1,x2,…,xp−12)∈Zpp−12\left(x_1,x_2,\dots,x_{\frac{p-1}2}\right)\in \mathbb{Z}_p^{\frac{p-1}2} such that ∑i=1p−12xi≡∑i=1p−12xi2≡⋯≡∑i=1p−12xip−12(modp).\sum_{i = 1}^{\frac{p-1}{2}} x_{i} \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{2} \equiv \cdots \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{\frac{p - 1}{2}} \pmod p.
Proposed by Ali Partofard
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