

A1. Prove that the sequence a_{0} = 2, 3, 6, 14, 40, 152, 784, ... with general term a_{n} = (n+4) a_{n1}  4n a_{n2} + (4n8) a_{n3} is the sum of two wellknown sequences.


A2. Can we find a subsequence of { n^{1/3}  m^{1/3} : n, m = 0, 1, 2, ... } which converges to √2?


A3. A convex pentagon has all its vertices lattice points in the plane (and no three collinear). Prove that its area is at least 5/2.


A4. Given a point P in the plane, let S_{P} be the set of points whose distance from P is irrational. What is the smallest number of such sets whose union is the entire plane?


A5. M and N are n x n matrices such that (MN)^{2} = 0. Must (NM)^{2} = 0?


A6. How many ordered pairs (A, B) of subsets of {1, 2, ... , 10} can we find such that each element of A is larger than B and each element of B is larger than A.


B1. R is the real line. Find all possible functions f: R → R with continuous derivative such that f(α)^{2} = 1990 + ∫_{0}^{α} ( f(x)^{2} + f '(x)^{2}) dx for all α.


B2. Let P_{n}(x, z) = ∏_{1}^{n} (1  z x^{i1}) / (z  x^{i}). Prove that 1 + ∑_{1}^{∞} (1 + x^{n}) P_{n}(x, z) = 0 for z > 1 and x < 1.


B3. Let S be the set of 2 x 2 matrices each of whose elements is one of the 15 squares 0, 1, 4, ... , 196. Prove that if we select more than 15^{4}  15^{2}  15 + 2 matrices from S, then two of those selected must commute.


B4. A finite group with n elements is generated by g and h. Can we arrange two copies of the elements of the group in a sequence (total length 2n) so that each element is g or h times the previous element and the first element is g or h times the last?


B5. Can we find a sequence of reals α_{i} ≠ 0 such that each polynomial α_{0} + α_{1}x + ... + α_{n}x^{n} has all its roots real and distinct?


B6. C is a nonempty, closed, bounded, convex subset of the plane. Given a support line L of C and a real number 0 ≤ α ≤ 1, let B_{α} be the band parallel to L, situated midway between L and the parallel support line on the other side of C, and of width α times the distance between the two support lines. What is the smallest α such that ∩ B_{α} contains a point of C, where the intersection is taken over all possible directions for the support line L?

