

A1. Show that we cannot have 4 binomial coefficients nCm, nC(m+1), nC(m+2), nC(m+3) with n, m > 0 (and m + 3 ≤ n) in arithmetic progression.


A2. Let S be a set with a binary operation * such that (1) a * (a * b) = b for all a, b ∈ S, (2) (a * b) * b = a for all a, b ∈ S. Show that * is commutative. Give an example for which S is not associative.


A3. A sequence { x_{i} } is said to have a Cesaro limit iff lim_{n→∞}(x_{1} + x_{2} + ... + x_{n})/n exists. Find all (realvalued) functions f on the closed interval [0, 1] such that { f(x_{i}) } has a Cesaro limit iff { x_{i} } has a Cesaro limit.


A4. Show that a circle inscribed in a square has a larger perimeter than any other ellipse inscribed in the square.


A5. Show that n does not divide 2^{n}  1 for n > 1.


A6. f is an integrable realvalued function on the closed interval [0, 1] such that ∫_{0}^{1} x^{m}f(x) dx = 0 for m = 0, 1, 2, ... , n  1, and 1 for m = n. Show that f(x) ≥ 2^{n}(n + 1) on a set of positive measure.


B1. Let ∑_{0}^{∞} x^{n}(x  1)^{2n} / n! = ∑_{0}^{∞} a_{n} x^{n}. Show that no three consecutive a_{n} are zero.


B2. A particle moves in a straight line with monotonically decreasing acceleration. It starts from rest and has velocity v a distance d from the start. What is the maximum time it could have taken to travel the distance d?


B3. A group has elements g, h satisfying: ghg = hg^{2}h, g^{3} = 1, h^{n} = 1 for some odd n. Prove h = 1.


B4. Show that for n > 1 we can find a polynomial p(a, b, c) with integer coefficients such that p(x^{n}, x^{n+1}, x + x^{n+2}) ≡ x.


B5. A, B, C and D are noncoplanar points. ∠ABC = ∠ADC and ∠BAD = ∠BCD. Show that AB = CD and BC = AD.


B6. The polynomial p(x) has all coefficients 0 or 1, and p(0) = 1. Show that if the complex number z is a root, then z ≥ (√5  1)/2.

