

A1. Four planar curves are defined as follows: C_{1} = {(x, y): x^{2}  y^{2} = x/(x^{2} + y^{2}) }, C_{2} = {(x, y): 2xy + y/(x^{2} + y^{2}) = 3}, C_{3} = {(x, y): x^{3}  3xy^{2} + 3y = 1}, C_{4} = {(x, y): 3yx^{2}  3x  y^{3} = 0}. Prove that C_{1} ∩ C_{2} = C_{3} ∩ C_{4}.


A2. An infinite sequence of decimal digits is obtained by writing the positive integers in order: 1234567891011 ... . Define f(n) = m if the 10^{n} th digit forms part of an mdigit number. For example, f(1) = 2, because the 10th digit is part of 10, and f(2) = 2, because the 100th digit is part of 55. Find f(1987).


A3. y = f(x) is a realvalued solution (for all real x) of the differential equation y''  2y' + y = 2e^{x} which is positive for all x. Is f '(x) necessarily positive for all x? y = g(x) is another real valued solution, which satisfies g'(x) > 0 for all real x. Is g(x) necessarily positive for all x?


A4. p(x, y, z) is a polynomial with real coefficients such that: (1) p(tx, ty, tz) = t^{2}f(y  x, z  x) for all real x, y, z, t (and some function f); (2) p(1, 0, 0) = 4, p(0 ,1, 0) = 5, and p(0, 0, 1) = 6; and (3) p(α, β, γ) = 0 for some complex numbers α, β, γ such that β  α = 10. Find γ  α.


A5. f: R^{2}→R^{3} (where R is the real line) is defined by f(x, y) = (y/(x^{2} + 4y^{2}), x/(x^{2} + 4y^{2}), 0). Can we find F: R^{3}→R^{3}, such that:
(1) if F = (F_{1}, F_{2}, F_{3}), then F_{i} all have continuous partial derivatives for all (x, y, z) ≠ (0, 0, 0);
(2) ∇ x F = 0 for all (x, y, z) ≠ 0;
(3) F(x, y, 0) = f(x, y)?


A6. Define f(n) as the number of zeros in the base 3 representation of the positive integer n. For which positive real x does F(x) = x^{f(1)}/1^{3} + x^{f(2)}/2^{3} + ... + x^{f(n)}/n^{3} + ... converge?


B1. Evaluate ∫_{2}^{4} ln^{1/2}(9  x)/( ln^{1/2}(9  x) + ln^{1/2}(x + 3) ) dx.


B2. Let n, r, s be nonnegative integers with n ≥ r + s, prove that ∑_{i=0}^{s} sCi / nC(r+i) = (n+1)/( (n+1s) (ns)Cr ), where mCn denotes the binomial coefficient.


B3. F is a field in which 1 + 1 ≠ 0. Define P_{α} = ( (α^{2}  1)/(α^{2} + 1), 2α/(α^{2} + 1) ). Let A = { (β, γ) : β, γ ∈ F, and β^{2} + γ^{2} = 1}, and let B= { (1, 0) } ∪ { P_{α}: α ∈ F, and α^{2} ≠ 1}. Prove that A = B.


B4. Define the sequences x_{i} and y_{i} as follows. Let (x_{1}, y_{1}) = (0.8, 0.6) and let (x_{n+1}, y_{n+1}) = (x_{n}cos y_{n}  y_{n}sin y_{n}, x_{n}sin y_{n} + y_{n} cos y_{n}) for n ≥ 1. Find lim_{n→∞} x_{n} and lim_{n→∞} y_{n}.


B5. A is a complex 2n x n matrix such that if z is a real 1 x 2n row vector then z A ≠ 0 unless z = 0. Prove that given any real 2n x 1 column vector x we can always find an n x 1 column vector z such that the real part of A z = x.


B6. F is a finite field with p^{2} elements, where p is an odd prime. S is a set of (p^{2}  1)/2 distinct nonzero elements of F such that for each a ∈ F, just one of a and a is in S. Prove that the number of elements in S ∩ {2a: a ∈ S} is even.

