

A1. Let Z be the integers. Prove that if f : Z → Z satisifies f( f(n) ) = f( f(n+2) + 2 ) = n for all n, and f(0) = 1, then f(n) = 1  n.


A2. Let the coefficient of x^{1992} in the power series (1 + x)^{α} = 1 + αx + ... be C(α). Find ∫_{0}^{1} C(y1) ∑_{k=1}^{1992}1/(y+k) dy.


A3. Find all positive integers a, b, m, n with m relatively prime to n such that (a^{2} + b^{2})^{m} = (ab)^{n}.


A4. Let R be the reals. Let f : R → R be an infinitely differentiable function such that f(1/n) = n^{2}/(n^{2}+1) for n = 1, 2, 3, ... Find the value of the derivatives of f at zero: f^{(k)}(0) for k = 1, 2, 3, ... .


A5. Let N be the positive integers. Define f : N → {0, 1} by f(n) = 1 if the number of 1s in the binary representation of n is odd and 0 otherwise. Show that there do not exist positive integers k and m such that f(k + j) = f(k + m + j) = f(k + 2m + j) for 0 ≤ j < m.


A6. Four points are chosen independently and at random on the surface of a sphere (using the uniform distribution). What is the probability that the center of the sphere lies inside the resulting tetrahedron?


B1. Let R be the reals. Let S ⊆ R have n ≥ 2 elements. Let A_{S} = { x ∈ R : x = (s + t)/2 for some s, t ∈ S with s ≠t}. What is the smallest possible A_{S}?


B2. Show that the coefficient of x^{k} in the expansion of (1 + x + x^{2} + x^{3})^{n} is ∑_{j=0}^{k} nCj nC(k2j).


B3. Let S be the set of points (x, y) in the plane such that the sequence a_{n} defined by a_{0} = x, a_{n+1} = (a_{n}^{2} + y^{2})/2 converges. What is the area of S?


B4. p(x) is a polynomial of degree < 1992 such that p(0), p(1), p(1) are all nonzero. The 1992th derivative of p(x)/(x^{3}  x) = f(x)/g(x) for polynomials f(x) and g(x). Find the smallest possible degree of f(x).


B5. Let A_{n} denote the n1 x n1 matrix (a_{ij}) with a_{ij} = i + 2 for i = j, and 1 otherwise. Is the sequence (det A_{n})/n! bounded?


B6. Let M be a set of real n x n matrices such that: (1) 1 ∈ M; (2) if A, B ∈ M, then just one of AB,  AB is in M; (3) if A, B ∈ M, then either AB = BA or AB = BA; (4) if 1 ≠ A ∈ M, then there is at least one B ∈ M such that AB =  BA. Prove that M contains at most n^{2} matrices.

