

A1. Show that the real polynomial ∑_{0}^{n} a_{i}x^{i} has at least one real root if ∑ a_{i}/(i + 1) = 0.


A2. A rough sphere radius R rests on top of a fixed rough sphere radius R. It is displaced slightly and starts to roll off. At what point does it lose contact?


A3. A sequence of numbers α_{i} ∈ [0, 1] is chosen at random. Show that the expected value of n, where ∑_{1}^{n} α_{i} > 1, ∑_{1}^{n1} α_{i} ≤ 1 is e.


A4. z_{1}, z_{2}, ... , z_{n} are complex numbers with modulus a > 0. Let f(n, m) denote the sum of all products of m of the numbers. For example, f(3, 2) = z_{1}z_{2} + z_{2}z_{3} + z_{3}z_{1}. Show that f(n, m) /a^{m}= f(n, nm) /a^{nm}.


A5. Let R be the reals. Show that there is at most one continuous function f : [0, 1]^{2} → R satisfying f(x, y) = 1 + ∫_{0}^{x}∫_{0}^{y} f(s, t) dt ds.


A6. Assume that the interest rate is r, so that capital of k becomes k(1 + r)^{n} after n years. How much do we need to invest to be able to withdraw 1 at the end of year 1, 4 at the end of year 2, 9 at the end of year 3, 16 at the end of year 4 and so on (in perpetuity)?


A7. Show that we cannot place 10 unit squares in the plane so that no two have an interior point in common and one has a point in common with each of the others.


B1. Do both (1) and (2):
(1) Given real numbers a, b, c, d with a > b, c, d, show how to construct a quadrilateral with sides a, b, c, d and the side length a parallel to that length b. What conditions must a, b, c, d satisfy?
(2) H is the foot of the altitude from A in the acuteangled triangle ABC. D is any point on the segment AH. BD meets AC at E, and CD meets AB at F. Show that ∠AHE = ∠AHF.


B2. Let n be a positive integer. Prove that n(n + 1)(n + 2)(n + 3) cannot be a square or a cube.


B3. In a tournament of n players, every pair of players plays once. There are no draws. Player i wins w_{i} games. Prove that we can find three players i, j, k such that i beats j, j beats k and k beats i iff ∑ w_{i}^{2} < (n  1)n(2n  1)/6.


B4. Let S be a spherical shell radius 1. Find the average straight line distance between two points of S. [In other words S is the set of points (x, y, z) with x^{2} + y^{2} + z^{2} = 1).


B5. S is an infinite set of points in the plane. The distance between any two points of S is integral. Prove that S is a subset of a straight line.


B6. A particle of unit mass moves in a vertical plane under the influence of constant gravitational force g and a resistive force which is in the opposite direction to its velocity and with magnitude a function of its speed. The particle starts at time t = 0 and has coordinates (x, y) at time t. Given that x = x(t) and is not constant, show that y(t) =  g x(t) ∫_{0}^{t} ds/x'(s) + g ∫_{0}^{t} x(s) / x'(s) ds + a x(t) + b, where a and b are constants.


B7. R is the reals. f : [a, b] → R is continuous and ∫_{a}^{b} x^{n}f(x) dx = 0 for all nonnegative integers n. Show that f(x) = 0 for all x.

