13th Vietnam 1975 problems

A1.  The roots of the equation x3 - x + 1 = 0 are a, b, c. Find a8 + b8 + c8.
A2.  Find all real x which satisfy (x3 + a3)/(x + a)3 + (x3 + b3)/(x + b)3 + (x3 + c3)/(x + c)3 + 3(x - a)(x - b)(x - c)/( 2(x + a)(x + b)(x + c) ) = 3/2.
A3.  ABCD is a tetrahedron. The three edges at B are mutually perpendicular. O is the midpoint of AB and K is the foot of the perpendicular from O to CD. Show that vol KOAC/vol KOBD = AC/BD iff 2·AC·BD = AB2.
B1.  Find all terms of the arithmetic progression -1, 18, 37, 56, ... whose only digit is 5.
B2.  Show that the sum of the maximum and minimum values of the function tan(3x)/tan3x on the interval (0, π/2) is rational.
B3.  L is a fixed line and A a fixed point not on L. L' is a variable line (in space) through A. Let M be the point on L and N the point on L' such that MN is perpendicular to L and L'. Find the locus of M and the locus of the midpoint of MN.

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