

A1. How many triples (A, B, C) are there of sets with A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A ∩ B ∩ C = ∅?


A2. ABC is an acuteangled triangle with area 1. A rectangle R has its vertices R_{i} on the sides of the triangle, R_{1} and R_{2} on BC, R_{3} on AC and R_{4} on AB. Another rectangle S has it vertices on the sides of the triangle AR_{3}R_{4}, two on R_{3}R_{4} and one one each of the other two sides. What is the maximum total area of R and S over all possible choices of triangle and rectangles?


A3. x is a real. Define a_{i 0} = x/2^{i}, a_{i j+1} = a_{i j}^{2} + 2 a_{i j}. What is lim_{n→∞} a_{n n}?


A4. Let a_{n} be the sequence defined by a_{1} = 3, a_{n+1} = 3^{k}, where k = a_{n}. Let b_{n} be the remainder when a_{n} is divided by 100. Which values b_{n} occur for infinitely many n?


A5. Let f_{n}(x) = cos x cos 2x ... cos nx. For which n in the range 1, 2, ... , 10 is ∫_{0}^{2π} f_{n}(x) dx nonzero?


A6. Find a polynomial f(x) with real coefficients and f(0) = 1, such that the sums of the squares of the coefficients of f(x)^{n} and (3x^{2} + 7x + 2)^{n} are the same for all positive integers n.


B1. p(x) is a polynomial of degree 5 with 5 distinct integral roots. What is the smallest number of nonzero coefficients it can have? Give a possible set of roots for a polynomial achieving this minimum.


B2. The polynomials p_{n}(x) are defined as follows: p_{0}(x) = 1; p_{n+1}'(x) = (n + 1) p_{n}(x + 1), p_{n+1}(0) = 0 for n ≥ 0. Factorize p_{100}(1) into distinct primes.


B3. a_{i j} is a positive integer for i, j = 1, 2, 3, ... and for each positive integer we can find exactly eight a_{i j} equal to it. Prove that a_{i j} > ij for some i, j.


B4. Let C be the circle radius 1, center the origin. A point P is chosen at random on the circumference of C, and another point Q is chosen at random in the interior of C. What is the probability that the rectangle with diagonal PQ, and sides parallel to the xaxis and yaxis, lies entirely inside (or on) C?


B5. Assuming that ∫_{∞}^{∞} e^{x2} dx = √π, find ∫_{0}^{∞} x^{1/2} e^{1985(x + 1/x)} dx.


B6. G is a finite group consisting of real n x n matrices with the operation of matrix multiplication. The sum of the traces of the elements of G is zero. Prove that the sum of the elements of G is the zero matrix.

