

A1. Dissect a regular 12gon into a regular hexagon, 6 squares and 6 equilateral triangles. Let the regular 12gon have vertices P_{1}, P_{2}, ... , P_{12} (in that order). Show that the diagonals P_{1}P_{9}, P_{12}P_{4} and P_{2}P_{11} are concurrent.


A2. The sequence a_{1}, a_{2}, a_{3}, ... of positive integers is strictly monotonic increasing, a_{2} = 2 and a_{mn} = a_{m}a_{n} for m, n relatively prime. Show that a_{n} = n.


A3. Let D be the differential operator d/dx and E the differential operator xD(xD  1)(xD  2) ... (xD  n). Find an expression of the form y = ∫_{1}^{x} g(t) dt for the solution of the differential equation Ey = f(x), with initial conditions y(1) = y'(1) = ... = y^{(n)}(1) = 0, where f(x) is a continuous realvalued function on the reals.


A4. Show that for any sequence of positive reals, a_{n}, we have lim sup_{n→∞} n( (a_{n+1} + 1)/a_{n}  1) ≥ 1. Show that we can find a sequence where equality holds.


A5. R is the reals. f : [0, π] → R is continuous and ∫_{0}^{π} f(x) sin x dx = ∫_{0}^{π} f(x) cos x dx = 0. Show that f is zero for at least two points in (0, π). Hence or otherwise, show that the centroid of any bounded convex open region of the plane is the midpoint of at least three distinct chords of its boundary.


A6. M is the midpoint of a chord PQ of an ellipse. A, B, C, D are four points on the ellipse such that AC and BD intersect at M. The lines AB and PQ meet at R, and the lines CD and PQ meet at S. Show that M is also the midpoint of RS.


B1. Find all integers n for which x^{2}  x + n divides x^{13} + x + 90.


B2. Is the set { 2^{m}3^{n}: m, n are integers } dense in the positive reals?


B3. R is the reals. Find all f : R → R which are twice differentiable and satisfy: f(x)^{2}  f(y)^{2} = f(x + y) f(x  y).


B4. Γ is a closed plane curve enclosing a convex region and having a continuously turning tangent. A, B, C are points of Γ such that ABC has the maximum possible perimeter p. Show that the normals to Γ at A, B, C are the angle bisectors of ABC. If A, B, C have this property, does ABC necessarily have perimeter p? What happens if Γ is a circle?


B5. The series ∑ a_{n} of nonnegative terms converges and a_{i} ≤ 100a_{n} for i = n, n + 1, n + 2, ... , 2n. Show that lim_{n→∞} na_{n} = 0.


B6. Let S = S_{0} be a set of points in space. Let S_{n} = { P : P belongs to the closed segment AB, for some A, B ∈ S_{n1}}. Show that S_{2} = S_{3}.

