VMO 1976

Find all integer solutions to $m^{m+n} = n^{12}, n^{m+n} = m^3$ .

Find all triangles $ABC$ such that $\frac{a cos A + b cos B + c cos C}{a sin A + b sin B + c sin C} =\frac{a + b + c}{9R}$ , where, as usual, $a, b, c$ are the lengths of sides $BC, CA, AB$ and $R$ is the circumradius.

$P$ is a point inside the triangle $ABC$ . The perpendicular distances from $P$ to the three sides have product $p$ . Show that $p \le \frac{ 8 S^3}{27abc}$ , where $S =$ area $ABC$ and $a, b, c$ are the sides. Prove a similar result for a tetrahedron.

Find all three digit integers $\overline{abc} = n$ , such that $\frac{2n}{3} = a! b! c!$

$L, L'$ are two skew lines in space and $p$ is a plane not containing either line. $M$ is a variable line parallel to $p$ which meets $L$ at $X$ and $L'$ at $Y$ . Find the position of $M$ which minimises the distance $XY$ . $L''$ is another fixed line. Find the line $M$ which is also perpendicular to $L''$ .

Show that $\frac{1}{x_1^n} + \frac{1}{x_2^n} +...+ \frac{1}{x_k^n} \ge k^{n+1}$ for positive real numbers $x_i$ with sum $1$ .