14th Vietnam 1976 problems

A1.  Find all integer solutions to mm+n = n12, nm+n = m3.
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 S3/(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/x1n + 1/x2n + ... + 1/xkn ≥ kn+1 for real numbers xi with sum 1.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.

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© John Scholes
23 July 2002