

A1. The roots of the equation x^{3}  x + 1 = 0 are a, b, c. Find a^{8} + b^{8} + c^{8}.


A2. Find all real x which satisfy (x^{3} + a^{3})/(x + a)^{3} + (x^{3} + b^{3})/(x + b)^{3} + (x^{3} + c^{3})/(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 = AB^{2}.

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)/tan^{3}x 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.
