Let k be a positive integer, let m\equal{}2^k\plus{}1, and let r=1 be a complex root of z^m\minus{}1\equal{}0. Prove that there exist polynomials P(z) and Q(z) with integer coefficients such that (P(r))^2\plus{}(Q(r))^2\equal{}\minus{}1. Putnamalgebrapolynomialcollege contests