2019 IMO Shortlist N5

Let $a$ be a positive integer. We say that a positive integer $b$ is $a$-good if $\binom{a n}{b}-1$ is divisible by $a n+1$ for all positive integers $n$ with $a n \geq b$. Suppose $b$ is a positive integer such that $b$ is $a$-good, but $b+2$ is not $a$-good. Prove that $b+1$ is prime. (Netherlands)

令 $a$ 为正整数。如果对于所有具有 $a n \geq b$ 的正整数 $n$，$\binom{a n}{b}-1$ 能被 $a n+1$ 整除，我们就说正整数 $b$ 是 $a$-good。假设 $b$ 是正整数，使得 $b$ 是 $a$-good，但 $b+2$ 不是 $a$-good。证明 $b+1$ 是素数。 (荷兰)