

A1. A circle has center on the side AB of the cyclic quadrilateral ABCD. The other three sides are tangent to the circle. Prove that AD + BC = AB.


A2. Let n and k be relatively prime positive integers with k < n. Each number in the set M = {1, 2, 3, ... , n1} is colored either blue or white. For each i in M, both i and ni have the same color. For each i in M not equal to k, both i and ik have the same color. Prove that all numbers in M must have the same color.


A3. For any polynomial P(x) = a_{0} + a_{1}x + ... + a_{k}x^{k} with integer coefficients, the number of odd coefficients is denoted by o(P). For i = 0, 1, 2, ... let Q_{i}(x) = (1 + x)^{i}. Prove that if i_{1}, i_{2}, ... , i_{n} are integers satisfying 0 ≤ i_{1} < i_{2} < ... < i_{n}, then:
o(Q_{i1} + Q_{i2} + ... + Q_{in}) ≥ o(Q_{i1}).


B1. Given a set M of 1985 distinct positive integers, none of which has a prime divisor greater than 23, prove that M contains a subset of 4 elements whose product is the 4th power of an integer.


B2. A circle center O passes through the vertices A and C of the triangle ABC and intersects the segments AB and BC again at distinct points K and N respectively. The circumcircles of ABC and KBN intersect at exactly two distinct points B and M. Prove that ∠OMB is a right angle.


B3. For every real number x_{1}, construct the sequence x_{1}, x_{2}, ... by setting:
x_{n+1} = x_{n}(x_{n} + 1/n).
Prove that there exists exactly one value of x_{1} which gives 0 < x_{n} < x_{n+1} < 1 for all n.

