Problem Solving Set 16

19 July 2012

1. Let n, k be positive integers and suppose that the polynomial x^{2k} −x^{k} +1 divides x^{2n}+x^{n}+1. Prove that x^{2k} +x^{k} +1 divides x^{2n} + x^{n} + 1.

2. Find all solutions in integers of

x^{3} + y^{3} + z^{3} = 3xyz.