

A1. A triangular number is a positive integer of the form n(n + 1)/2. Show that m is a sum of two triangular numbers iff 4m + 1 is a sum of two squares.


A2. For what region of the real (a, b) plane, do both (possibly complex) roots of the polynomial z^{2} + az + b = 0 satisfy z < 1?


A3. Let 0 < α < β < γ ∈ R, the reals. Let K = { (x, y, z) : x^{β} + y^{β} + z^{β} = 1, and x, y, z ≥ 0} ∈ R^{3}. Define f : K → R by f(x, y, z) = x^{α} + y^{β} + z^{γ}. At what points of K does f assume its maximum and minimum values?


A4. m > 1 is odd. Let n = 2m and θ = e^{2πi/n}. Find a finite set of integers {a_{i}} such that ∑ a_{i} θ^{i} = 1/(1  θ).


A5. Let I be an interval and f(x) a continuous realvalued function on I. Let y_{1} and y_{2} be linearly independent solutions of y'' = f(x) y, which take positive values on I. Show that from some positive constant k, k √(y_{1} y_{2}) is a solution of y'' + 1/y^{3} = f(x) y.


A6. Given three points in space forming an acuteangled triangle, show that we can find two further points such that no three of the five points are collinear and the line through any two is normal to the plane through the other three.


B1. Let G be the group { (m, n) : m, n are integers } with the operation (a, b) + (c, d) = (a + c, b + d). Let H be the smallest subgroup containing (3, 8), (4, 1) and (5, 4). Let H_{ab} be the smallest subgroup containing (0, a) and (1, b). Find a > 0 such that H_{ab} = H.


B2. A slab is the set of points strictly between two parallel planes. Prove that a countable sequence of slabs, the sum of whose thicknesses converges, cannot fill space.


B3. Let n be a fixed positive integer. Let S be any finite collection of at least n positive reals (not necessarily all distinct). Let f(S) = (∑_{a∈S} a)^{n}, and let g(S) = the sum of all nfold products of the elements of S (in other words, the nth symmetric function). Find sup_{S} g(S)/f(S).


B4. Does a circle have a subset which is topologically closed and which contains just one of each pair of diametrically opposite points?


B5. Define f_{0}(x) = e^{x}, f_{n+1}(x) = x f_{n}'(x). Show that ∑_{0}^{∞} f_{n}(1)/n! = e^{e}.


B6. Let h_{n} = ∑_{1}^{n} 1/r. Show that n  (n  1) n^{1/(n1)} > h_{n} > n(n + 1)^{1/n}  n for n > 2.

