IMO 1967 LL GBR17

Origin: GBR

Problem

Let k, m, and n be positive integers such that m+k + 1 is a prime number greater than n + 1. Write cs for s(s + 1). Prove that the product (cm+1 −ck)(cm+2 −ck) \cdot \cdot \cdot (cm+n −ck) is divisible by the product c1c2 \cdot \cdot \cdot cn.

Solution

Using cr −cs = (r −s)(r + s + 1) we can easily get (cm+1 −ck) \cdot \cdot \cdot (cm+n −ck) c1c2 \cdot \cdot \cdot cn = (m −k + n)! (m −k)!n! \cdot (m + k + n + 1)! (m + k + 1)!(n + 1)!. The first factor (m−k+n)! (m−k)!n! =
m−k+n n  is clearly an integer. The second factor is also an integer because by the assumption, m + k + 1 and (m + k)!(n + 1)! are coprime, and (m + k + n + 1)! is divisible by both; hence it is also divisible by their product.