32nd Vietnam 1994 problems

A1.  Find all real solutions to:
x3 + 3x - 3 + ln(x2 - x + 1) = y
y3 + 3y - 3 + ln(y2 - y + 1) = z
z3 + 3z - 3 + ln(z2 - z + 1) = x.
A2.  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.
A3.  Define the sequence x0, x1, x2, ... by x0 = a, where 0 < a < 1, xn+1 = 4/π2 (cos-1xn + π/2) sin-1xn. Show that the sequence converges and find its limit.
B1.  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.
B2.  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.
B3.  Do there exist polynomials p(x), q(x), r(x) whose coefficients are positive integers such that p(x) = (x2 - 3x + 3) q(x) and q(x) = (x2/20 - x/15 + 1/12) r(x)?

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