MathDB

N3

Part of 2015 IMO Shortlist

Problems(1)

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

Source: 2015 IMO Shortlist N3

7/7/2016
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.
IMO Shortlistnumber theoryDivisibility