

A1. Let A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6} be distinct points in the plane. Let D be the longest distance between any pair, and d the shortest distance. Show that D/d ≥ √3.


A2. α is a real number. Find all continuous realvalued functions f :[0, 1] → (0, ∞) such that ∫_{0}^{1} f(x) dx = 1, ∫_{0}^{1} x f(x) dx = α, ∫_{0}^{1} x^{2} f(x) dx = α^{2}.


A3. The distinct points x_{n} are dense in the interval (0, 1). x_{1}, x_{2}, ... , x_{n1} divide (0, 1) into n subintervals, one of which must contain x_{n}. This part is divided by x_{n} into two subintervals, lengths a_{n} and b_{n}. Prove that ∑ a_{n}b_{n}(a_{n} + b_{n}) = 1/3.


A4. The sequence of integers u_{n} is bounded and satisfies u_{n} = (u_{n1} + u_{n2} + u_{n3}u_{n4})/(u_{n1}u_{n2} + u_{n3} + u_{n4}). Show that it is periodic for sufficiently large n.


A5. Find a constant k such that for any positive a_{i}, ∑_{1}^{∞} n/(a_{1} + a_{2} + ... + a_{n}) ≤ k ∑_{1}^{∞}1/a_{n}.


A6. S is a finite set of collinear points. Let k be the maximum distance between any two points of S. Given a pair of points of S a distance d < k apart, we can find another pair of points of S also a distance d apart. Prove that if two pairs of points of S are distances a and b apart, then a/b is rational.


B1. a_{n} are positive integers such that ∑ 1/a_{n} converges. b_{n} is the number of a_{n} which are ≤ n. Prove lim b_{n}/n = 0.


B2. S is a finite set. A set P of subsets of S has the property that any two members of P have at least one element in common and that P cannot be extended (whilst keeping this property). Prove that P contains just half of the subsets of S.


B3. R is the reals. f : R → R is continuous and for any α > 0, lim_{n→∞} f(nα) = 0. Prove lim_{x→∞} f(x) = 0.


B4. n great circles on the sphere are in general position (in other words at most two circles pass through any two points on the sphere). How many regions do they divide the sphere into?


B5. Let a_{n} be a strictly monotonic increasing sequence of positive integers. Let b_{n} be the least common multiple of a_{1}, a_{2}, ... , a_{n}. Prove that ∑ 1/b_{n} converges.


B6. D is a disk. Show that we cannot find congruent sets A, B with A ∩ B = ∅, A ∪ B = D.

