2015 IMO Shortlist N3

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_{k}=(m+k) /(n+k)$ for $k=$ $1,2, \ldots, n+1$. Prove that if all the numbers $x_{1}, x_{2}, \ldots, x_{n+1}$ are integers, then $x_{1} x_{2} \cdots x_{n+1}-1$ is divisible by an odd prime. (Austria)

令 $m$ 和 $n$ 为正整数，使得 $m>n$。定义 $x_{k}=(m+k) /(n+k)$ 为 $k=$ $1,2, \ldots, n+1$。证明如果所有数字 $x_{1}, x_{2}, \ldots, x_{n+1}$ 都是整数，则 $x_{1} x_{2} \cdots x_{n+1}-1$ 可以被奇素数整除。 (奥地利)