Write a full solution.
Theorems allowed to use: basic inequalities, AM-GM-HM, Cauchy Schwarz, Holder, Weighted AM-GM, Jensen, Power Mean, Chebyshev, Rearrangement, Weirstrass, Bernoulli
Comments by professor: These 2 problems can be solved using AM-GM and Cauchy Schwarz only. Maybe you can try it!
1.) Let be sides of triangle, , prove that for all positive integers
2.) Let such that , prove that
1.) Find all functions that satisfy for all .
2.) Find all strictly monotonic functions that satisfy for all .
This note is part of Thailand Math POSN 3rd round 2015