

A1. The set of pairs of positive reals (x, y) such that x^{y} = y^{x} form the straight line y = x and a curve. Find the point at which the curve cuts the line.


A2. For which real numbers α, β can we find a constant k such that x^{α}y^{β} < k(x + y) for all positive x, y?


A3. Find lim_{n→∞} ∑_{1}^{N} n/(N + i^{2}), where N = n^{2}.


A4. If n = ∏p^{r} be the prime factorization of n, let f(n) = (1)^{∑ r} and let F(n) = ∑_{dn} f(d). Show that F(n) = 0 or 1. For which n is F(n) = 1?


A5. Let X be a set of n points. Let P be a set of subsets of X, such that if A, B ∈ P, then X  A, A ∪ B, A ∩ B ∈ P. What are the possible values for the number of elements of P?


A6. Consider polynomials in one variable over the finite field F_{2} with 2 elements. Show that if n + 1 is not prime, then 1 + x + x^{2} + ... + x^{n} is reducible. Can it be reducible if n + 1 is prime?


A7. S is a nonempty closed subset of the plane. The disk (a circle and its interior) D ⊇ S and if any disk D' ⊇ S, then D' ⊇ D. Show that if P belongs to the interior of D, then we can find two distinct points Q, R ∈ S such that P is the midpoint of QR.


B1. a_{n} is a sequence of positive reals. h = lim (a_{1} + a_{2} + ... + a_{n})/n and k = lim (1/a_{1} + 1/a_{2} + ... + 1/a_{n})/n exist. Show that h k ≥ 1.


B2. Two points are selected independently and at random from a segment length β. What is the probability that they are at least a distance α (< β) apart?


B3. A, B, C, D lie in a plane. No three are collinear and the four points do not lie on a circle. Show that one point lies inside the circle through the other three.


B4. Given x_{1}, x_{2}, ... , x_{n} ∈ [0, 1], let s = ∑_{1≤i<j≤n} x_{i}  x_{j}. Find f(n), the maximum value of s over all possible {x_{i}}.


B5. Let n be an integer greater than 2. Define the sequence a_{m} by a_{1} = n, a_{m+1} = n to the power of a_{m}. Either show that a_{m} < n!! ... ! (where the factorial is taken m times), or show that a_{m} > n!! ... ! (where the factorial is taken m1 times).


B6. Let y be the solution of the differential equation y'' =  (1 + √x) y such that y(0) = 1, y'(0) = 0. Show that y has exactly one zero for x ∈ (0, π/2) and find a positive lower bound for it.


B7. The sequence of nonnegative reals satisfies a_{n+m} ≤ a_{n}a_{m} for all m, n. Show that lim a_{n}^{1/n} exists.

