Question bank VMO 1962

Prove that for all positive real numbers $a,b,c,d$,
$$\frac{1}{\frac1a+\frac1b}+\frac{1}{\frac1c+\frac1d}\le\frac{1}{\frac1{a+c}+\frac1{b+d}}.$$

Find the first derivative at $x=1$ of
$$f(x)=(1+x)\cdot\sqrt{2+x^2}\cdot\sqrt[3]{3+x^3}.$$

Let $ABCD$ be a tetrahedron, and let $A',B'$ be the orthogonal projections of $A,B$ onto their opposite face planes. Prove that the lines $AA'$ and $BB'$ intersect if and only if $AB$ and $CD$ are perpendicular. If $AC=AD=BC=BD$, do $AA'$ and $BB'$ intersect?

A regular square pyramid has height $h$. Each lateral face makes an angle $\alpha$ with the base. Through a base edge construct a section perpendicular to the opposite lateral face. Find the volume of the pyramid thereby formed and discuss the formula obtained.

Solve over the real numbers:
$$\sin^6x+\cos^6x=\frac14.$$