

A1. e^{bx} cos cx is expanded in a Taylor series ∑ a_{n} x^{n}. b and c are positive reals. Show that either all a_{n} are nonzero, or infinitely many a_{n} are zero.


A2. p(x, y) = a x^{2} + b x y + c y^{2} is a homogeneous real polynomial of degree 2 such that b^{2} < 4ac, and q(x, y) is a homogeneous real polynomial of degree 3. Show that we can find k > 0 such that p(x, y) = q(x, y) has no roots in the disk x^{2} + y^{2} < k except (0, 0).


A3. A perfect square has length n if its last n digits (in base 10) are the same and nonzero. What is the longest possible length? What is the smallest square achieving this length?


A4. The real sequence a_{1}, a_{2}, a_{3}, ... has the property that lim_{n→∞} (a_{n+2}  a_{n}) = 0. Prove that lim_{n→∞} (a_{n+1}  a_{n})/n = 0.


A5. Find the radius of the largest circle on an ellipsoid with semiaxes a > b > c.


A6. x is chosen at random from the interval [0, a] (with the uniform distribution). y is chosen similarly from [0, b], and z from [0, c]. The three numbers are chosen independently, and a ≥ b ≥ c. Find the expected value of min(x, y, z).


B1. Let f(n) = (n^{2} + 1)(n^{2} + 4)(n^{2} + 9) ... (n^{2} + (2n)^{2}). Find lim_{n→∞} f(n)^{1/n}/n^{4}.


B2. A weather station measures the temperature T continuously. It is found that on any given day T = p(t), where p is a polynomial of degree <= 3, and t is the time. Show that we can find times t_{1} < t_{2}, which are independent of p, such that the average temperature over the period 9am to 3pm is ( p(t_{1}) + p(t_{2}) / 2. Show that t_{1} ≈ 10:16am, t_{2} ≈ 1:44pm.


B3. S is a closed subset of the real plane. Its projection onto the xaxis is bounded. Show that its projection onto the yaxis is closed.


B4. A vehicle covers a mile (= 5280 ft) in a minute, starting and ending at rest and never exceeding 90 miles/hour. Show that its acceleration or deceleration exceeded 6.6 ft/sec^{2}.


B5. k_{n}(x) = n on (∞, n], x on [n, n], and n on [n, ∞). Prove that the (real valued) function f(x) is continuous iff all k_{n}( f(x) ) are continuous.


B6. The quadrilateral Q contains a circle which touches each side. It has side lengths a, b, c, d and area √(abcd). Prove it is cyclic.

