VMO 1994

Find all real solutions to: $x^3$ + 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.

ABC is a triangle. Reflect each vertex in the opposite side to get the triangle A'B'C'. Find a necessary and sufficient condition on ABC for A'B'C' to be equilateral.

Define the sequence $x_0$, $x_1$, $x_2$, ... by $x_0$ = a, where 0 < a < 1, $x_{n+1}$ = 4/π$^2$ ($cos^{-1}x_n$ + π/2) $sin^{-1}x_n$. Show that the sequence converges and find its limit.

There are n+1 containers arranged in a circle. One container has n stones, the others are empty. A move is to choose two containers A and B, take a stone from A and put it in one of the containers adjacent to B, and to take a stone from B and put it in one of the containers adjacent to A. We can take A = B. For which n is it possible by series of moves to end up with one stone in each container except that which originally held n stones.

S is a sphere center O. G and G' are two perpendicular great circles on S. Take A, B, C on G and D on G' such that the altitudes of the tetrahedron ABCD intersect at a point. Find the locus of the intersection.

Do there exist polynomials p(x), q(x), r(x) whose coefficients are positive integers such that p(x) = ($x^2$ - 3x + 3) q(x) and q(x) = ($x^2$/20 - x/15 + 1/12) r(x)?