Problem

Source: 2011 Saudi Arabia BMO TST 4.4 - Balkan MO

Tags: number theory, remainder



Let $p \ge 3$ be a prime. For $j = 1,2 ,... ,p - 1$, let $r_j$ be the remainder when the integer $\frac{j^{p-1}-1}{p}$ is divided by $p$. Prove that $$r_1 + 2r_2 + ... + (p - 1)r_{p-1} \equiv \frac{p+1}{2} (\mod p)$$