A1. 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.
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 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.
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) = (x^{2} - 3x + 3) q(x) and q(x) = (x^{2}/20 - x/15 + 1/12) r(x) ? |