

A1. Prove that ∫_{0}^{1} x^{4}(1  x)^{4}/(1 + x^{2}) dx = 22/7  π.


A2. Given integers a, b, c, d such that ad  bc ≠ 0, integers m, n and a real ε > 0, show that we can find rationals x, y, such that 0 < ax + by  m < ε and 0 < cx + dy  n < ε.


A3. S is a finite set. P is the set of all subsets of S. Show that we can label the elements of P as A_{i}, such that A_{1} = ∅ and for each n ≥ 1, either A_{n1} ⊂ A_{n} and A_{n}  A_{n1} = 1, or A_{n1} ⊃ A_{n} and A_{n1}  A_{n} = 1.


A4. Let S_{2} be the 2sphere { (x, y, z) : x^{2} + y^{2} + z^{2} = 1}. Show that for any n points on S_{2}, the sum of the squares of the n(n  1)/2 distances between them (measured in space, not in S_{2}) is at most n^{2}.


A5. Find the smallest possible α such that if p(x) ≡ ax^{2} + bx + c satisfies p(x) ≤ 1 on [0, 1], then p'(0) ≤ α.


A6. Find all finite polynomials whose coefficients are all ±1 and whose roots are all real.


B1. The random variables X, Y can each take a finite number of integer values. They are not necessarily independent. Express prob( min(X, Y) = k) in terms of p_{1} = prob( X = k), p_{2} = prob(Y = k) and p_{3} = prob( max(X, Y) = k).


B2. (G, *) is a finite group with n elements. K is a subset of G with more than n/2 elements. Prove that for every g ∈ G, we can find h, k ∈ K such that g = h * k.


B3. Given that a 60^{o} angle cannot be trisected with ruler and compass, prove that a 120^{o}/n angle cannot be trisected with ruler and compass for n = 1, 2, 3, ... .


B4. R is the reals. f : R → R is continuous and L = ∫_{∞}^{∞} f(x) dx exists. Show that ∫_{∞}^{∞} f(x  1/x) dx = L.


B5. Let F be the field with p elements. Let S be the set of 2 x 2 matrices over F with trace 1 and determinant 0. Find S.


B6. A compact set of real numbers is closed and bounded. Show that we cannot find compact sets A_{1}, A_{2}, A_{3}, ... such that (1) all elements of A_{n} are rational and (2) given any compact set K whose members are all rationals, K ⊆ some A_{n}.

