

A1. a_{n} is a sequence of positive reals satisfying a_{n} ≤ a_{2n} + a_{2n+1} for all n. Prove that ∑ a_{n} diverges.


A2. Let R be the region in the first quadrant bounded by the xaxis, the line 2y = x, and the ellipse x^{2}/9 + y^{2} = 1. Let R' be the region in the first quadrant bounded by the yaxis, the line y = mx, and the ellipse. Find m such that R and R' have the same area.


A3. X is the set of points on one or more sides of a triangle with sides length 1, 1 and √2. Show that if X is 4colored, then there must be two points of the same color a distance 2  √2 or more apart.


A4. A and B are 2 x 2 matrices with integral values. A, A + B, A + 2B, A + 3B, and A + 4B all have inverses with integral values. Show that A + 5B does also.


A5. Given a sequence of positive real numbers which tends to zero, show that every nonempty interval (a, b) contains a nonempty subinterval (c, d) that does not contain any numbers equal to a sum of 1994 distinct elements of the sequence.


A6. Let Z be the integers. Let f_{1}, f_{2}, ... , f_{10} : Z → Z be bijections. Given any n ∈ Z we can find some composition of the f_{i} (allowing repetitions) which maps 0 to n. Consider the set of 1024 functions S = { g_{1}g_{2}... g_{10}}, where g_{i} = the identity or f_{i}. Show that at most half the functions in S map a finite (nonempty) subset of Z onto itself.


B1. For a positive integer n, let f(n) be the number of perfect squares d such that n  d ≤ 250. Find all n such that f(n) = 15. [The perfect squares are 0, 1, 4, 9, 16, ... .]


B2. For which real α does the curve y = x^{4} + 9x^{3} + α x^{2} + 9x + 4 contain four collinear points?


B3. Let R be the reals and R^{+} the positive reals. f : R → R^{+} is differentiable and f '(x) > f(x) for all x. For what k must f(x) exceed e^{kx} for all sufficiently large x?


B4. A is the 2 x 2 matrix (a_{ij}) with a_{11} = a_{22} = 3, a_{12} = 2, a_{21} = 4 and I is the 2 x 2 unit matrix. Show that the greatest common divisor of the entries of A^{n}  I tends to infinity.


B5. For any real α define f_{α}(x) = [αx]. Let n be a positive integer. Show that there exists an α such that for 1 ≤ k ≤ n, f_{α}^{k}(n^{2}) = n^{2}  k = f_{αk}(n^{2}), where f_{α}^{k} denotes the kfold composition of f_{α}.


B6. a, b, c, d are integers in the range 0  99. Show that if 101a 100·2^{a} + 101b  100·2^{b} = 101c  100·2^{c} + 101d  100·2^{d} (mod 10100) then {a, b} = {c, d}.

