

A1. C is the complex numbers. f : C → R is defined by f(z) = z^{3}  z + 2. What is the maximum value of f on the unit circle z = 1?


A2. K is a cone. s is a sphere radius r, and S is a sphere radius R. s is inside K touches it along all points of a circle. S is also inside K and touches it along all points of a circle. s and S also touch each other. What is the volume of the finite region between the two spheres and inside K?


A3. a_{n} is a sequence of positive reals decreasing monotonically to zero. b_{n} is defined by b_{n} = a_{n}  2a_{n+1} + a_{n+2} and all b_{n} are nonnegative. Prove that b_{1} + 2b_{2} + 3b_{3} + ... = a_{1}.


A4. Let D be a disk radius r. Given (x, y) ∈ D, and R > 0, let a(x, y, R) be the length of the arc of the circle center (x, y), radius R, which is outside D. Evaluate lim_{R→0} R^{2} ∫_{D} a(x, y, R) dx dy.


A5. Let α_{1}, α_{2}, ... , α_{n} be the nth roots of unity. Find ∏_{i<j} (α_{i}  α_{j})^{2}.


A6. Do either (1) or (2):
(1) On each element ds of a closed plane curve there is a force 1/R ds, where R is the radius of curvature. The force is towards the center of curvature at each point. Show that the curve is in equilibrium.
(2) Prove that x + 2/3 x^{3} + 2.4/3.5 x^{5} + ... + 2.4. ... .2n/(3.5. ... .2n+1) x^{2n+1} + ... = (1  x^{2})^{1/2} sin^{1}x


B1. p(x) is a cubic polynomial with roots α, β, γ and p'(x) divides p(2x). Find the ratios α : β : γ.


B2. A circle radius r is tangent to the three coordinate planes (x =0, y =0, z = 0) in space. Find the locus of its center.

B3. Show that [√n + √(n + 1)] = [√(4n + 2)] for positive integers n.


B4. R is the reals. For what λ can we find a continuous function f : (0, 1) → R, not identically zero, such that ∫_{0}^{1} min(x, y) f(y) dy = λ f(x) for all x ∈ (0, 1)?

B5. Find the area of the region { (x, y) : x + yt + t^{2} ≤ 1 for all t ∈ [0, 1] }.


B6. Do either (1) or (2):
(1) Take the origin O of the complex plane to be the vertex of a cube, so that OA, OB, OC are edges of the cube. Let the feet of the perpendiculars from A, B, C to the complex plane be the complex numbers u, v, w. Show that u^{2} + v^{2} + w^{2} = 0.
(2) Let (a_{ij}) be an n x n matrix. Suppose that for each i, 2 a_{ii} > ∑_{1}^{n} a_{ij}. By considering the corresponding system of linear equations or otherwise, show that det a_{ij} ≠ 0.

