ILL 1971 Problem 3

Let $a, b, c$ be positive real numbers, $0 < a \leq b \leq c$ . Prove that for any positive real numbers $x, y, z$ the following inequality holds: $$(ax+by+cz) \left( \frac xa + \frac yb+\frac zc \right) \leq (x+y+z)^2 \cdot \frac{(a+c)^2}{4ac}.$$