

A1. p(x) is a real polynomial of degree less than 3 and satisfies p(x) ≤ 1 for x ∈ [1, 1]. Show that p'(x) ≤ 4 for x ∈ [1, 1].


A2. R is the reals. For functions f, g : R → R and x ∈ R define I(fg) = ∫_{1}^{x} f(t) g(t) dt. If a(x), b(x), c(x), d(x) are real polynomials, show that I(ac) I(bd)  I(ad) I(bc) is divisible by (x  1)^{4}.


A3. ABCD are the vertices of a square with A opposite C and side AB = s. The distances of a point P in space from A, B, C, D are a, b, c, d respectively. Show that a^{2} + c^{2} = b^{2} + d^{2}, and that the perpendicular distance k of P from the plane ABCD is given by 8k^{2} = 2(a^{2} + b^{2} + c^{2} + d^{2})  4s^{2}  (a^{4} + b^{4} + c^{4} + d^{4}  2a^{2}c^{2}  2b^{2}d^{2})/s^{2}.


A4. R is the reals. f : R → R has a continuous derivative, f(0) = 0, and f '(x) ≤ f(x) for all x. Show that f is constant.


A5. Let T be a tangent plane to the ellipsoid x^{2}/a^{2} + y^{2}/b^{2} + z^{2}/c^{2} = 1. What is the smallest possible volume for the tetrahedral volume bounded by T and the planes x = 0, y = 0, z = 0?


A6. A particle moves in one dimension. Its distance x from the origin at time t is at + bt^{2} + ct^{3}. Find an expression for the particle's acceleration in terms of a, b, c and its speed v.


B1. Two circles C_{1} and C_{2} intersect at A and B. C_{1} has radius 1. L denotes the arc AB of C_{2} which lies inside C_{1}. L divides C_{1} into two parts of equal area. Show L has length > 2.


B2. P_{0} is the parabola y^{2} = mx, vertex K (0, 0). If A and B points on P_{0} whose tangents are at right angles, let C be the centroid of the triangle ABK. Show that the locus of C is a parabola P_{1}. Repeat the process to define P_{n}. Find the equation of P_{n}.


B3. The density of a solid sphere depends solely on the radial distance. The gravitational force at any point inside the sphere, a distance r from the center, is kr^{2} (where k is a constant). Find the density (in terms of G, k and r), and the gravitational force at a point outside the sphere. [You may assume the usual results about the gravitational attraction of a spherical shell.]


B4. Define a_{n} = 2(1 + 1/n)^{2n+1}/( (1 + 1/n)^{n} + (1 + 1/n)^{n+1}). Prove that a_{n} is strictly monotonic increasing.


B5. Let m be the smallest integer greater than (√3 + 1)^{2n}. Show that m is divisible by 2^{n+1}.


B6. The particle P moves in the plane. At t = 0 it starts from the point A with velocity zero. It is next at rest at t = T, when its position is the point B. Its path from A to B is the arc of a circle center O. Prove that its acceleration at each point in the time interval [0, T] is nonzero, and that at some point in the interval its acceleration is directly towards the center O.

