

A1. Let d be any positive integer not equal to 2, 5 or 13. Show that one can find distinct a, b in the set {2, 5, 13, d} such that ab  1 is not a perfect square.


A2. Given a point P_{0} in the plane of the triangle A_{1}A_{2}A_{3}. Define A_{s} = A_{s3} for all s >= 4. Construct a set of points P_{1}, P_{2}, P_{3}, ... such that P_{k+1} is the image of P_{k} under a rotation center A_{k+1} through an angle 120^{o} clockwise for k = 0, 1, 2, ... . Prove that if P_{1986} = P_{0}, then the triangle A_{1}A_{2}A_{3} is equilateral.


A3. To each vertex of a regular pentagon an integer is assigned, so that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers x, y, z respectively, and y < 0, then the following operation is allowed: x, y, z are replaced by x + y, y, z + y respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.


B1. Let A, B be adjacent vertices of a regular ngon (n ≥ 5) with center O. A triangle XYZ, which is congruent to and initially coincides with OAB, moves in the plane in such a way that Y and Z each trace out the whole boundary of the polygon, with X remaining inside the polygon. Find the locus of X.


B2. Find all functions f defined on the nonnegative reals and taking nonnegative real values such that: f(2) = 0, f(x) ≠ 0 for 0 ≤ x < 2, and f(xf(y)) f(y) = f(x + y) for all x, y.


B3. Given a finite set of points in the plane, each with integer coordinates, is it always possible to color the points red or white so that for any straight line L parallel to one of the coordinate axes the difference (in absolute value) between the numbers of white and red points on L is not greater than 1?

