VMO 1962 Problem 1

Prove that for positive real numbers $a$ , $b$ , $c$ , $d$ , we have $$\frac{1}{\frac{1}{a}+\frac{1}{b}}+\frac{1}{\frac{1}{c}+\frac{1}{d}}\le\frac{1}{\frac{1}{a+c}+\frac{1}{b+d}}$$