21st Putnam 1960 Problem A5 The real polynomial p(x) is such that for any real polynomial q(x), we have p(q(x)) = q(p(x)). Find p(x).
Take q(x) = x + k and set x = 0. Then we have p(k) = p(0) + k. This is true for all k, so p(x) must be x + c for some c. Now take q(x) = x 2 . We get x 2 + c = (x + c) 2 = x 2 + 2cx + c 2 . Hence c = 0 and p(x) = x. Comment: it is even easier to take q(x) = k. Then p(k) = k for all k. In other words, p(x) = x. 21st Putnam 1960 © John Scholes jscholes@kalva.demon.co.uk 15 Feb 2002