VMO 1995

Find all real solutions to $x^3$ - $3x^2$ - 8x + 40 = 8(4x + 4)$^{1/4}$ = 0.

The sequence $a_0$, $a_1$, $a_2$, ... is defined by $a_0$ = 1, $a_1$ = 3, $a_{n+2}$ = $a_{n+1}$ + $9a_n$ for n even, $9a_{n+1}$ + $5a_n$ for n odd. Show that $a_{1995}^2$ + $a_{1996}^2$ + $a_{1997}^2$ + $a_{1998}^2$ + $a_{1999}^2$ + $a_{2000}^2$ is divisible by 20, and that no $a_{2n+1}$ is a square.

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.

ABCD is a tetrahedron. A' is the circumcenter of the face opposite A. B', C', D' are defined similarly. $p_A$ is the plane through A perpendicular to C'D', $p_B$ is the plane through B perpendicular to D'A', $p_C$ is the plane through C perpendicular to A'B', and $p_D$ 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?

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.

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.