Problem Solving Set 18

21 July 2012

1. Let n be a positive integer. Prove that 2^{n−1} divides

∑_{0≤k<n/2} \binom{n}{2k + 1} 5^{k}.

2. Find all ordered triples of primes (p, q, r) such that p | q^{r} + 1, q | r^{p} + 1, r | p^{q} + 1.