Thursday, Jan 12th

Problem
Let $x,y,z$ be positive reals such that $x^2+y^2+z^2=1$. Prove that $x^2yz+xy^2z+xyz^2\leq 1/3$

Solution
Notice that $x^2yz+xy^2z+xyz^2=xyz(x+y+z)$. Then using the AM-GM inequality together with the AM-RMS inequality gives the desired result.