33rd Vietnam 1995 problems

A1.  Find all real solutions to x3 - 3x2 - 8x + 40 = 8(4x + 4)1/4 = 0.
A2.  The sequence a0, a1, a2, ... is defined by a0 = 1, a1 = 3, an+2 = an+1 + 9an for n even, 9an+1 + 5an for n odd. Show that a19952 + a19962 + a19972 + a19982 + a19992 + a20002 is divisible by 20, and that no a2n+1 is a square.
A3.  ABC is a triangle with altitudes AD, BE, CF. A', B', C' are points on AD, BE, CF such that AA'/AD = BB'/BE = CC'/CF = k. Find all k such that A'B'C' is similar to ABC for all triangles ABC.
B1.  ABCD is a tetrahedron. A' is the circumcenter of the face opposite A. B', C', D' are defined similarly. pA is the plane through A perpendicular to C'D', pB is the plane through B perpendicular to D'A', pC is the plane through C perpendicular to A'B', and pD is the plane through D perpendicular to B'C'. Show that the four planes have a common point. If this point is the circumcenter of ABCD, must ABCD be regular?
B2.  Find all real polynomials p(x) such that p(x) = a has more than 1995 real roots, all greater than 1995, for any a > 1995. Multiple roots are counted according to their multiplicities.
B3.  How many ways are there of coloring the vertices of a regular 2n-gon with n colors, such that each vertex is given one color, and every color is used for two non-adjacent vertices? Colorings are regarded as the same if one is obtained from the other by rotation.

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

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© John Scholes
22 July 2002