

A1. k is a positive constant. The sequence x_{i} of positive reals has sum k. What are the possible values for the sum of x_{i}^{2} ?


A2. Show that we can find infinitely many triples N, N + 1, N + 2 such that each member of the triple is a sum of one or two squares.


A3. An octagon is incribed in a circle. One set of alternate vertices forms a square area 5. The other set forms a rectangle area 4. What is the maximum possible area for the octagon?


A4. Show that lim_{k→∞} &inf;_{0}^{k} sin x sin x^{2} dx converges.


A5. A, B, C each have integral coordinates and lie on a circle radius R. Show that at least one of the distances AB, BC, CA exceeds R^{1/3}.


A6. p(x) is a polynomial with integer coefficients. A sequence x_{0}, x_{1}, x_{2}, ... is defined by x_{0} = 0, x_{n+1} = p(x_{n}). Prove that if x_{n} = 0 for some n > 0, then x_{1} = 0 or x_{2} = 0.


B1. We are given N triples of integers (a_{1}, b_{1}, c_{1}), (a_{2}, b_{2}, c_{2}), ... (a_{N}, b_{N}, c_{N}). At least one member of each triple is odd. Show that we can find integers A, B, C such that at least 4N/7 of the N values A a_{i} + B b_{i} + C c_{i} are odd.


B2. m and n are positive integers with m ≤ n. d is their greatest common divisor. nCm is the binomial coefficient. Show that d/n nCm is integral.


B3. a_{1}, a_{2}, ... , a_{N} are real and a_{N} is nonzero. f(x) = a_{1} sin 2πx + a_{2} sin 4πx + a_{3} sin 6πx + ... + a_{N} sin 2Nπx. Show that the number of zeros of f^{(i)}(x) = 0 in the interval [0, 1) is a nondecreasing function of i and tends to 2N (as i tends to infinity).


B4. f(x) is a continuous real function satisfying f(2x^{2}  1) = 2 x f(x). Show that f(x) is zero on the interval [1, 1].


B5. S_{0} is an arbitrary finite set of positive integers. Define S_{n+1} as the set of integers k such that just one of k  1, k is in S_{n}. Show that for infinitely many n, S_{n} is the union of S_{0} and a translate of S_{0}.


B6. Let X be the set of 2^{n} points (±1, ±1, ... , ±1) in Euclidean nspace. Show that any subset of X with at least 2^{n+1}/n points contains an equilateral triangle.

