

A1. We are given a positive integer n and real numbers a_{i} such that ∑_{1}^{n} a_{k} sin kx ≤ sin x for all real x. Prove ∑_{1}^{n} k a_{k} ≤ 1.


A2. Let u_{n} be the number of symmetric n x n matrices whose elements are all 0 or 1, with exactly one 1 in each row. Take u_{0} = 1. Prove u_{n+1} = u_{n} + n u_{n1} and ∑_{0}^{∞} u_{n} x^{n}/n! = e^{f(x)}, where f(x) = x + (1/2) x^{2}.


A3. Find the smallest positive integer n such that we can find a polynomial nx^{2} + ax + b with integer coefficients and two distinct roots in the interval (0, 1).


A4. Let 1/2 < α ∈ R, the reals. Show that there is no function f : [0, 1] → R such that f(x) = 1 + α ∫_{x}^{1} f(t) f(t  x) dt for all x ∈ [0, 1].


A5. K is a convex, finite or infinite, region of the plane, whose boundary is a union of a finite number of straight line segments. Its area is at least π/4. Show that we can find points P, Q in K such that PQ = 1.


A6. a_{i} and b_{i} are reals such that a_{1}b_{2} ≠ a_{2}b_{1}. What is the maximum number of possible 4tuples (sign x_{1}, sign x_{2}, sign x_{3}, sign x_{4}) for which all x_{i} are nonzero and x_{i} is a simultaneous solution of a_{1}x_{1} + a_{2}x_{2} + a_{3}x_{3} + a_{4}x_{4} = 0 and b_{1}x_{1} + b_{2}x_{2} + b_{3}x_{3} + b_{4}x_{4} = 0. Find necessary and sufficient conditions on a_{i} and b_{i} for this maximum to be achieved.


B1. A hexagon is inscribed in a circle radius 1. Alternate sides have length 1. Show that the midpoints of the other three sides form an equilateral triangle.


B2. α, β ∈ [0, 1] and we have ax^{2} + bxy + cy^{2} ≡ (αx + (1  α)y)^{2}, (αx + (1  α)y)(βx + (1  β)y) ≡ dx^{2} + exy + fy^{2}. Show that at least one of a, b, c ≥ 4/9 and at least one of d, e, f ≥ 4/9.


B3. R is the reals. f, g are continuous functions R → R with period 1. Show that lim_{n→∞} ∫_{0}^{1} f(x) g(nx) dx = (∫_{0}^{1} f(x) dx) (∫_{0}^{1} g(x) dx).


B4. We are given a sequence a_{1}, a_{2}, ... , a_{n}. Each a_{i} can take the values 0 or 1. Initially, all a_{i} = 0. We now successively carry out steps 1, 2, ... , n. At step m we change the value of a_{i} for those i which are a multiple of m. Show that after step n, a_{i} = 1 iff i is a square. Devise a similar scheme to give a_{i} = 1 iff i is twice a square.


B5. The first n terms of the exansion of (2  1)^{n} are 2^{n} ( 1 + n/1! (1/2) + n(n + 1)/2! (1/2)^{2} + ... + n(n + 1) ... (2n 2)/(n  1)! (1/2)^{n1} ). Show that they sum to 1/2.


B6. R is the reals. D is the closed unit disk x^{2} + y^{2} ≤ 1 in R^{2}. The function f : D → [1, 1] has partial derivatives f_{1}(x, y) and f_{2}(x, y). Show that there is a point (a, b) in the interior of D such that f_{1}(a, b)^{2} + f_{2}(a, b)^{2} ≤ 16.

