

A1. Find all integer solutions to m^{m+n} = n^{12}, n^{m+n} = m^{3}.


A2. Find all triangles ABC such that (a cos A + b cos B + c cos C)/(a sin A + b sin B + c sin C) = (a + b + c)/9R, where, as usual, a, b, c are the lengths of sides BC, CA, AB and R is the circumradius.

A3. P is a point inside the triangle ABC. The perpendicular distances from P to the three sides have product p. Show that p ≤ 8 S^{3}/(27abc), where S = area ABC and a, b, c are the sides. Prove a similar result for a tetrahedron.

B1. Find all three digit integers abc = n, such that 2n/3 = a! b! c!


B2. 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".

B3. Show that 1/x_{1}^{n} + 1/x_{2}^{n} + ... + 1/x_{k}^{n} ≥ k^{n+1} for real numbers x_{i} with sum 1.
