Problem

Source: VII International Festival of Young Mathematicians Sozopol, 2016 Theme for 10-12 grade

Tags: algebra, inequalities



Let $x\leq y\leq z$ be real numbers such that $x+y+z=12$, $x^2+y^2+z^2=54$. Prove that: a) $x\leq 3$ and $z\geq 5$ b) $xy$, $yz$, $zx\in [9,25]$