VMO 1962

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

Let $f(x) = (1 + x)\cdot\sqrt{(2 + x^2)}\cdot\sqrt[3]{(3 + x^3)}$ . Determine $f'(1)$ .

Let $ABCD$ is a tetrahedron. Denote by $A'$ , $B'$ the feet of the perpendiculars from $A$ and $B$ , respectively to the opposite faces. Show that $AA'$ and $BB'$ intersect if and only if $AB$ is perpendicular to $CD$ . Do they intersect if $AC = AD = BC = BD$ ?

Let be given a tetrahedron $ABCD$ such that triangle $BCD$ equilateral and $AB = AC = AD$ . The height is $h$ and the angle between two planes $ABC$ and $BCD$ is $\alpha$ . The point $X$ is taken on $AB$ such that the plane $XCD$ is perpendicular to $AB$ . Find the volume of the tetrahedron $XBCD$ .

Solve the equation $\sin^6x + \cos^6x = \frac{1}{4}$ .