VMO 1994 — Division A

Solve the system
$$\begin{cases}x^3+3x-3+\ln(x^2-x+1)=y,\\ y^3+3y-3+\ln(y^2-y+1)=z,\\ z^3+3z-3+\ln(z^2-z+1)=x.\end{cases}$$

Let $ABC$ be a triangle. Reflect $A,B,C$ in the lines containing their opposite sides to obtain $A',B',C'$, respectively. Determine a necessary and sufficient condition on $ABC$ for the triangles $ABC$ and $A'B'C'$ to have equal areas.

Define $x_0,x_1,x_2,\ldots$ by $x_0=a$, where $0\lt a\lt 1$, and
$$x_{n+1}=\frac4{\pi^2}\left(\arccos x_n+\frac\pi2\right)\arcsin x_n.$$
Prove that the sequence converges and determine its limit.

There are $n+1$ boxes arranged on a circle. One box contains $n$ stones; the others are empty. A move is defined as follows: choose two boxes $A$ and $B$, take two stones from $A$, put one into a box adjacent to $B$, and put the other into a box adjacent to $A$. Note that $A=B$ is allowed. For which $n$ can a sequence of such moves result in each stone being in a separate box?

Let $S$ be a sphere with center $O$, and let $G,G'$ be two great circles whose planes are perpendicular. Choose $A,B,C$ on $G$ and $D$ on $G'$ so that the four altitudes of the nondegenerate tetrahedron $ABCD$ are concurrent. Determine the locus of their common point.

Do there exist polynomials $p(x),q(x),r(x)$ whose coefficients are positive integers, satisfying
$$p(x)=(x^2-3x+3)q(x),\qquad q(x)=\left(\frac{x^2}{20}-\frac{x}{15}+\frac1{12}\right)r(x)?$$