IMO 1982 LL AUS1

Origin: AUS

Problem

It is well known that the binomial coefficients
n k 

n! k!(n−k)!, 0 \leqk \leqn, are positive integers. The factorial n! is defined inductively by 0! = 1, n! = n \cdot (n −1)! for n \geq1. (a) Prove that n+1
2n n  is an integer for n \geq0. (b) Given a positive integer k, determine the smallest integer Ck with the property that Ck n+k+1
2n n+k  is an integer for all n \geqk.