IrMO 2012 Problem 5

(a) Show that if $x$ and $y$ are positive real numbers, then $$(x+y)^5\ge 12xy(x^3+y^3)$$ (b) Prove that the constant $12$ is the best possible. In other words, prove that for any $K>12$ there exist positive real numbers $x$ and $y$ such that $$(x+y)^5<Kxy(x^3+y^3)$$