VMO 1963

A conference has $47$ people attending. One woman knows $16$ of the men who are attending, another knows $17$ , and so on up to the $m$ -th woman who knows all $n$ of the men who are attending. Find $m$ and $n$ .

For what values of $m$ does the equation $x^2 + (2m + 6)x + 4m + 12 = 0$ has two real roots, both of them greater than $-1$ .

Solve the equation $\sin^3x \cos 3x + \cos^3x \sin 3x = \frac{3}{8}$ .

The tetrahedron $S.ABC$ has the faces $SBC$ and $ABC$ perpendicular. The three angles at $S$ are all $60^{\circ}$ and $SB = SC = 1$ . Find the volume of the tetrahedron.

The triangle $ABC$ has semiperimeter $p$ . Find the side length $BC$ and the area $S$ in terms of $\angle A$ , $\angle B$ and $p$ . In particular, find $S$ if $p \approx 23.6$ , $\angle A \approx 52^{\circ}42'$ , $\angle B \approx 46^{\circ}16'$ .