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Irreduced to ashes, modulo $p$

Source: Kyiv City MO 2021 Round 1, Problem 8.5

December 21, 2023
number theoryprime numbers

Problem Statement

For a prime number p>3p > 3, define the following irreducible fraction:
mn=p−12+p−23+…+2p−1−1\frac{m}{n} = \frac{p-1}{2} + \frac{p-2}{3} + \ldots + \frac{2}{p-1} - 1
Prove that mm is divisible by pp.
Proposed by Oleksii Masalitin
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