

A1. A surface S in 3space is such that every normal intersects a fixed line L. Show that we can find a surface of revolution containing S.


A2. k is a real number greater than 1. A uniform wire consists of the curve y = e^{x} between x = 0 and x = k, and the horizontal line y = e^{k} between x = k  1 and x = k. The wire is suspended from (k  1, e^{k}) and a horizontal force applied at the other end, (0, 1) to keep it in equilibrium. Show that the force is directed towards increasing x.


A3. A and B are real numbers such that cos A ≠ cos B. Show that for any integer n > 1, cos nA cos B  cos A cos nB < (n^{2}  1) cos A  cos B.


A4. p(z) is a polynomial of degree n with complex coefficients. Its roots (in the complex plane) can be covered by a disk radius r. Show that for any complex k, the roots of n p(z)  k p'(z) can be covered by a disk radius r + k.


A5. Let S be a set of n points in the plane such that the greatest distance between two points of S is 1. Show that at most n pairs of points of S are a distance 1 apart.


A6. Define a_{n} by a_{1} = ln α, a_{2} = ln(α  a_{1}), a_{n+1} = a_{n} + ln(α  a_{n}). Show that lim_{n→∞} a_{n} = α  1.


A7. Show that we can find a set of disjoint circles such that given any rational point on the xaxis, there is a circle touching the xaxis at that point. Show that we cannot find such a set for the irrational points.


B1. Let A be the 100 x 100 matrix with a_{mn} = mn. Show that the absolute value of each of the 100! products in the expansion of det A is congruent to 1 mod 101.


B2. The sequence a_{n} is defined by its initial value a_{1}, and a_{n+1} = a_{n}(2  k a_{n}). For what real a_{1} does the sequence converge to 1/k?


B3. R^{+} is the positive reals, f : [0, 1] → R^{+} is monotonic decreasing. Show that ∫_{0}^{1} f(x) dx ∫_{0}^{1} x f(x)^{2} dx ≤ ∫_{0}^{1} x f(x) dx ∫_{0}^{1} f(x)^{2} dx.


B4. Show that the number of ways of representing n as an ordered sum of 1s and 2s equals the number of ways of representing n + 2 as an ordered sum of integers > 1. For example: 4 = 1 + 1 + 1 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 2 + 1 = 1 + 1 + 2 (5 ways) and 6 = 4 + 2 = 2 + 4 = 3 + 3 = 2 + 2 + 2 (5 ways).


B5. Let S be a set and P the set of all subsets of S. f : P → P is such that if X ⊆ Y, then f(X) ⊆ f(Y). Show that for some K, f(K) = K.


B6. y is the solution of the differential equation (x^{2} + 9) y'' + (x^{2} + 4)y = 0, y(0) = 0, y'(0) = 1. Show that y(x) = 0 for some x ∈ [√(63/53)π, 3π/2].


B7. Let P be a regular polygon and its interior. Show that for any n > 1, we can find a subset S_{n} of the plane such that we cannot translate and rotate P to cover S_{n} but we can translate and rotate P to cover any n points of S_{n}.

