

A1. Find polynomials a(x), b(x), c(x) such that a(x)  b(x) + c(x) = 1 for x < 1, 3x + 2 for 1 ≤ x ≤ 0, 2x + 2 for x > 0.


A2. Show that for some fixed positive n we can always express a polynomial with real coefficients which is nowhere negative as a sum of the squares of n polynomials.


A3. Let 1/(1  2x  x^{2}) = s_{0} + s_{1}x + s_{2}x^{2} + ... . Prove that for some f(n) we have s_{n}^{2} + s_{n+1}^{2} = s_{f(n)}.


A4. Let a_{ij} = i^{2}j/(3^{i}(j 3^{i} + i 3^{j})). Find ∑ a_{ij} where the sum is taken over all pairs of integers (i, j) with i, j > 0.


A5. Find a constant k such that for any polynomial f(x) of degree 1999, we have f(0) ≤ k ∫_{1}^{1} f(x) dx.


A6. u_{1} = 1, u_{2} = 2, u_{3} = 24, u_{n} = (6 u_{n1}^{2}u_{n3}  8 u_{n1}u_{n2}^{2})/(u_{n2}u_{n3}). Show that u_{n} is always a multiple of n.


B1. The triangle ABC has AC = 1, ∠ACB = 90^{o}, and ∠BAC = φ. D is the point between A and B such that AD = 1. E is the point between B and C such that ∠EDC = φ. The perpendicular to BC at E meets AB at F. Find lim_{φ→0} EF.


B2. p(x) is a polynomial of degree n. q(x) is a polynomial of degree 2. p(x) = p''(x)q(x) and the roots of p(x) are not all equal. Show that the roots of p(x) are all distinct.


B3. Let R be the reals. Define f :[0, 1) x [0, 1) → R by f(x, y) = ∑ x^{m}y^{n}, where the sum is taken over all pairs of positive integers (m, n) satisfying m ≥ n/2, n ≥ m/2. Find lim_{(x, y)→(1, 1)} (1  xy^{2})(1  x^{2}y)f(x, y).


B4. Let R be the reals. f :R → R is three times differentiable, and f(x), f '(x), f ''(x), f '''(x) are all positive for all x. Also f(x) ≥ f '''(x) for all x. Show that f '(x) < 2 f(x) for all x.


B5. n is an integer greater than 2 and φ = 2π/n. A is the n x n matrix (a_{ij}), where a_{ij} = cos( (i + j)φ) for i ≠ j, 1 + cos(2 j φ) for i = j. Find det A.


B6. X is a finite set of integers greater than 1 such that for any positive integer n, we can find m ∈ X such that m divides n or is relatively prime to n. Show that X contains a prime or two elements whose greatest common divisor is prime.

