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Found problems: 1

2022 IMC, 3
Let $p$ be a prime number. A flea is staying at point $0$ of the real line. At each minute, the flea has three possibilities: to stay at its position, or to move by $1$ to the left or to the right. After $p-1$ minutes, it wants to be at $0$ again. Denote by $f(p)$ the number of its strategies to do this (for example, $f(3) = 3$: it may either stay at $0$ for the entire time, or go to the left and then to the right, or go to the right and then to the left). Find $f(p)$ modulo $p$.
function generating functions counting techniques modular arithmetic number theory