32nd Vietnamese Mathematical Olympiad 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, x_n+1 = 4/π² (cos^-1x_n + π/2) sin^-1x_n. 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)?

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