Let $x, y, z$ be nonnegative real numbers with $xy + yz + zx = 1$. Show that: $$\frac{4}{x + y + z} \le (x + y)(\sqrt3 z + 1).$$
Source: Switzerland - 2017 Swiss MO Final Round p10
Tags: algebra, inequalities
Let $x, y, z$ be nonnegative real numbers with $xy + yz + zx = 1$. Show that: $$\frac{4}{x + y + z} \le (x + y)(\sqrt3 z + 1).$$