

A1. a and b are positive reals and a > b. Let C be the plane curve r = a  b cos θ. For what values of b/a is C convex?


A2. Does the series ∑_{2}^{∞} 1/ln n! converge? Does the series 1/3 + 1/(3 3^{1/2}) + 1/(3 3^{1/2} 3^{1/3}) + ... + 1/(3 3^{1/2} 3^{1/3} ... 3^{1/n}) + ... converge?


A3. The sequence a_{n} is defined by a_{0} = α, a_{1} = β, a_{n+1} = a_{n} + (a_{n1}  a_{n})/(2n). Find lim a_{n}.


A4. Do either (1) or (2)
(1) P is a prism with triangular base. A is a vertex. The total area of the three faces containing A is 3k. Show that if the volume of P is maximized, then each of the three faces has area k and the two lateral faces are perpendicular to each other.
(2) Let f(x) = x + x^{3}/(1·3) + x^{5}/(1·3·5) + x^{7}/(1·3·5·7) + ... , and g(x) = 1 + x^{2}/2 + x^{4}/(2·4) + x^{6}/(2·4·6) + ... . Show that ∫_{0}^{x} exp(  t^{2}/2) dt = f(x)/g(x).


A5. Let N be the set of natural numbers {1, 2, 3, ... }. Let Z be the integers. Define d : N → Z by d(1) = 0, d(p) = 1 for p prime, and d(mn) = m d(n) + n d(m) for any integers m, n. Determine d(n) in terms of the prime factors of n. Find all n such that d(n) = n. Define d_{1}(m) = d(m) and d_{n+1}(m) = d(d_{n}(m)). Find lim_{n→∞} d_{n}(63).


A6. Let f(x) = ∑_{0}^{∞} a_{n}x^{n} and suppose that each a_{n} = 0 or 1. Do either (1) or (2):
(1) Show that if f(1/2) is rational, then f(x) has the form p(x)/q(x) for some integer polynomials p(x) and q(x).
(2) Show that if f(1/2) is not rational, then f(x) does not have the form p(x)/q(x) for any integer polynomials p(x) and q(x).


B1. Given n, not necessarily distinct, points P_{1}, P_{2}, ... , P_{n} on a line. Find the point P on the line to minimize ∑ PP_{i}.


B2. An ellipse with semiaxes a and b has perimeter length p(a, b). For b/a near 1, is π(a + b) or 2π√(ab) the better approximation to p(a, b)?


B3. Leap years have 366 days; other years have 365 days. Year n > 0 is a leap year iff (1) 4 divides n, but 100 does not divide n, or (2) 400 divides n. n is chosen at random from the natural numbers. Show that the probability that December 25 in year n is a Wednesday is not 1/7.


B4. A long, light cylinder has elliptical crosssection with semiaxes a > b. It lies on the ground with its main axis horizontal and the major axes horizontal. A thin heavy wire of the same length as the cylinder is attached to the line along the top of the cylinder. [We could take the cylinder to be the surface x ≤ L, y^{2}/a^{2} + z^{2}/b^{2} = 1. Contact with the ground is along x ≤ L, y = 0, z = b. The wire is along x ≤ L, y = 0, z = b.] For what values of b/a is the cylinder in stable equilibrium?


B5. Do either (1) or (2):
(1) Show that if ∑(a_{n} + 2a_{n+1}) converges, then so does ∑ a_{n}.
(2) Let S be the surface 2xy = z^{2}. The surface S and the variable plane P enclose a cone with volume πa^{3}/3, where a is a positive real constant. Find the equation of the envelope of P. What is the envelope in the case of a general cone?


B6.
(1) The convex polygon C' lies inside the polygon C. Is it true that the perimeter of C' is no longer than the perimeter of C?
(2) C is the convex polygon with shortest perimeter enclosing the polygon C'. Is it true that the perimeter of C is no longer than the perimeter of C' ?
(3) The closed convex surface S' lies inside the closed surface S. Is it true that area S' ≤ area S?
(4) S is the smallest convex surface containing the closed surface S'. Is it true that area S ≤ area S'?

