

A1. Let S = {1, 4, 7, 10, 13, 16, ... , 100}. Let T be a subset of 20 elements of S. Show that we can find two distinct elements of T with sum 104.


A2. Let A be the real n x n matrix (a_{ij}) where a_{ij} = a for i < j, b (≠ a) for i > j, and c_{i} for i = j. Show that det A = (b p(a)  a p(b) )/(b  a), where p(x) = ∏ (c_{i}  x).


A3. Let p(x) = 2(x^{6} + 1) + 4(x^{5} + x) + 3(x^{4} + x^{2}) + 5x^{3}. Let a = ∫_{0}^{∞} x/p(x) dx, b = ∫_{0}^{∞} x^{2}/p(x) dx, c = ∫_{0}^{∞} x^{3}/p(x) dx, d = ∫_{0}^{∞} x^{4}/p(x) dx. Which of a, b, c, d is the smallest?


A4. A binary operation (represented by multiplication) on S has the property that (ab)(cd) = ad for all a, b, c, d. Show that: (1) if ab = c, then cc = c; (2) if ab = c, then ad = cd for all d. Find a set S, and such a binary operation, which also satisfies: (A) a a = a for all a; (B) ab = a ≠ b for some a, b; (C) ab ≠ a for some a, b.


A5. Let a_{1}, a_{2}, ... , a_{n} be reals in the interval (0, π) with arithmetic mean μ. Show that ∏ (sin a_{i})/a_{i} ≤ ( (sin μ)/μ )^{n}.


A6. Given n points in the plane, prove that less than 2n^{3/2} pairs of points are a distance 1 apart.


B1. A convex octagon inscribed in a circle has 4 consecutive sides length 3 and the remaining sides length 2. Find its area.


B2. Find ∑_{1}^{∞}∑_{1}^{∞} 1/(i^{2}j + 2ij + ij^{2}).


B3. The polynomials p_{n}(x) are defined by p_{1}(x) = 1 + x, p_{2}(x) = 1 + 2x, p_{2n+1}(x) = p_{2n}(x) + (n + 1) x p_{2n1}(x), p_{2n+2}(x) = p_{2n+1}(x) + (n + 1) x p_{2n}(x). Let a_{n} be the largest real root of p_{n}(x). Prove that a_{n} is monotonic increasing and tends to zero.


B4. Show that we can find integers a, b, c, d such that a^{2} + b^{2} + c^{2} + d^{2} = abc + abd + acd + bcd, and the smallest of a, b, c, d is arbitarily large.


B5. Find the real polynomial p(x) of degree 4 with largest possible coefficient of x^{4} such that p( [1, 1] ) ⊆ [0, 1].


B6. a_{ij} are reals in [0, 1]. Show that ( ∑_{i=1}^{n} ∑_{j=1}^{mi} a_{ij}/i )^{2} ≤ 2m ∑_{i=1}^{n} ∑_{j=1}^{mi} a_{ij}.

