RU-ALLRUSSIAN 1961 Problem 5

(a) Given a quadruple (a, b, c, d) of positive reals, transform to the new quadruple (ab, bc, cd, da). Repeat arbitarily many times. Prove that you can never return to the original quadruple unless a=b=c=d=1. (b) Given n a power of 2, and an n-tuple (a₁, a₂, ..., aₙ) transform to a new n-tuple (a₁a₂, a₂a₃, ..., aₙ-₁aₙ, aₙa₁). If all the members of the original n-tuple are 1 or -1, prove that with sufficiently many repetitions you obtain all 1s.