

A1. How many positive integers divide at least one of 10^{40} and 20^{30}?


A2. A clock's minute hand has length 4 and its hour hand length 3. What is the distance between the tips at the moment when it is increasing most rapidly?


A3. Let f(n) = 1 + 2n + 3n^{2} + ... + (p  1)n^{p2}, where p is an odd prime. Prove that if f(m) = f(n) (mod p), then m = n (mod p).


A4. Prove that for m = 5 (mod 6), mC2  mC5 + mC8  mC11 + ...  mC(m6) + mC(m3) ≠ 0.


A5. Does there exist a positive real number α such that [α^{n}]  n is even for all integers n > 0?


A6. Let T be the triangle with vertices (0, 0), (a, 0), and (0, a). Find lim_{a→∞} a^{4}exp(a^{3}) ∫_{T} exp(x^{3}+y^{3}) dx dy.


B1. Let C be a cube side 4, center O. Let S be the sphere center O radius 2. Let A be one of the vertices of the cube. Let R be the set of points in C but not S, which are closer to A than to any other vertex of C. Find the volume of R.


B2. Let f(n) be the number of ways of representing n as a sum of powers of 2 with no power being used more than 3 times. For example, f(7) = 4 (the representations are 4 + 2 + 1, 4 + 1 + 1 + 1, 2 + 2 + 2 + 1, 2 + 2 + 1 + 1 + 1). Can we find a real polynomial p(x) such that f(n) = [p(n)] ?


B3. y_{1}, y_{2}, y_{3} are solutions of y''' + a(x) y'' + b(x) y' + c(x) y = 0 such that y_{1}^{2} + y_{2}^{2} + y_{3}^{2} = 1 for all x. Find constants α, β such that y_{1}'(x)^{2} + y_{2}'(x)^{2} + y_{3}'(x)^{2} is a solution of y' + α a(x) y + βc(x) = 0.


B4. Let f(n) = n + [√n]. Define the sequence a_{i} by a_{0} = m, a_{n+1} = f(a_{n}). Prove that it contains at least one square.


B5. Define x as the distance from x to the nearest integer. Find lim_{n→∞} 1/n ∫_{1}^{n} n/x dx. You may assume that ∏_{1}^{∞} 2n/(2n1) 2n/(2n+1) = π/2.


B6. Let α be a complex (2^{n} + 1)th root of unity. Prove that there always exist polynomials p(x), q(x) with integer coefficients, such that p(α)^{2} + q(α)^{2} = 1.

