VMO 1995 — Division A

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

The integer sequence $\{a_n\}_{n=0}^{+\infty}$ is defined by $a_0=1$, $a_1=3$, and
$$a_{n+2}=\begin{cases}a_{n+1}+9a_n,&n\text{ even},\\ 9a_{n+1}+5a_n,&n\text{ odd}.\end{cases}$$
Prove 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 term $a_{2m+1}$ is a perfect square.

Let $ABC$ be a triangle with altitudes $AD,BE,CF$. Points $A',B',C'$ lie on $AD,BE,CF$, respectively, and satisfy
$$\frac{AA'}{AD}=\frac{BB'}{BE}=\frac{CC'}{CF}=k.$$
Determine all $k$ for which triangle $A'B'C'$ is similar to triangle $ABC$.

Let $ABCD$ be a tetrahedron. Let $A'$ be the circumcenter of the face opposite $A$, and define $B',C',D'$ similarly. Let $p_A$ be the plane through $A$ perpendicular to $C'D'$, $p_B$ the plane through $B$ perpendicular to $D'A'$, $p_C$ the plane through $C$ perpendicular to $A'B'$, and $p_D$ the plane through $D$ perpendicular to $B'C'$. Prove that these four planes have a common point. If this common point coincides with the circumcenter of tetrahedron $ABCD$, must the tetrahedron be regular? Explain.

Determine all polynomials $p(x)$ such that, for every $a\gt 1995$, the equation $p(x)=a$ has more than $1995$ real roots, all greater than $1995$ (a root of multiplicity greater than one is counted according to that multiplicity).

How many ways are there to color the vertices of a regular $2n$-gon with $n$ colors, so that each vertex receives one color and each color is used on two nonadjacent vertices? Two colorings are considered the same if one is obtained from the other by a rotation.