For a prime number $p > 3$, define the following irreducible fraction: $$\frac{m}{n} = \frac{p-1}{2} + \frac{p-2}{3} + \ldots + \frac{2}{p-1} - 1$$ Prove that $m$ is divisible by $p$. Proposed by Oleksii Masalitin
Source: Kyiv City MO 2021 Round 1, Problem 8.5
Tags: number theory, prime numbers
For a prime number $p > 3$, define the following irreducible fraction: $$\frac{m}{n} = \frac{p-1}{2} + \frac{p-2}{3} + \ldots + \frac{2}{p-1} - 1$$ Prove that $m$ is divisible by $p$. Proposed by Oleksii Masalitin