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x_1x_2...x_(n+1)-1 is divisible by an odd prime

Source: 2015 IMO Shortlist N3

July 7, 2016
IMO Shortlistnumber theoryDivisibility

Problem Statement

Let mm and nn be positive integers such that m>nm>n. Define xk=m+kn+kx_k=\frac{m+k}{n+k} for k=1,2,…,n+1k=1,2,\ldots,n+1. Prove that if all the numbers x1,x2,…,xn+1x_1,x_2,\ldots,x_{n+1} are integers, then x1x2…xn+1−1x_1x_2\ldots x_{n+1}-1 is divisible by an odd prime.
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