7th CanMO 1975 Problem 8 Find all real polynomials p(x) such that p(p(x) ) = p(x) n for some positive integer n.
Let p(x) have highest term a x m . Then the highest term in p(p(x) ) is a m+1 x M , where M = m 2 , and the highest term in p(x) n is a n x mn . Hence m = n and a = 1. But that means p(p(x) ) = p(x) n + the other terms, so the other terms must all be zero. Hence p(x) = x n . 7th CanMO 1975 © John Scholes jscholes@kalva.demon.co.uk 10 June 2002