

A1. Let A, B, C, D be four distinct points on a line, in that order. The circles with diameter AC and BD intersect at X and Y. The line XY meets BC at Z. Let P be a point on the line XY other than Z. The line CP intersects the circle with diameter AC at C and M, and the line BP intersects the circle with diameter BD at B and N. Prove that the lines AM, DN, XY are concurrent.


A2. Let a, b, c be positive real numbers with abc = 1. Prove that:
1/(a^{3}(b + c)) + 1/(b^{3}(c + a)) + 1/(c^{3}(a + b)) ≥ 3/2.


A3. Determine all integers n > 3 for which there exist n points A_{1}, ... , A_{n} in the plane, no three collinear, and real numbers r_{1}, ... , r_{n} such that for any distinct i, j, k, the area of the triangle A_{i}A_{j}A_{k} is r_{i} + r_{j} + r_{k}.


B1. Find the maximum value of x_{0} for which there exists a sequence x_{0}, x_{1}, ... , x_{1995} of positive reals with x_{0} = x_{1995} such that for i = 1, ... , 1995:
x_{i1} + 2/x_{i1} = 2x_{i} + 1/x_{i}.


B2. Let ABCDEF be a convex hexagon with AB = BC = CD and DE = EF = FA, such that ∠BCD = ∠EFA = 60^{o}. Suppose that G and H are points in the interior of the hexagon such that ∠AGB = ∠DHE = 120^{o}. Prove that AG + GB + GH + DH + HE ≥ CF.


B3. Let p be an odd prime number. How many pelement subsets A of {1, 2, ... , 2p} are there, the sum of whose elements is divisible by p?

