

A1. Do either (1) or (2)
(1) Let L be the line through (0, a, a) parallel to the xaxis, M the line through (a, 0, a) parallel to the yaxis, and N the line through (a, a, 0) parallel to the zaxis. Find the equation of S, the surface formed from the union of all lines K which intersect each of L, M and N.
(2) Let S be the surface xy + yz + zx = 0. Which planes cut S in circles? In parabolas?

A2. Take points O, P, Q, R in space. Let the volume of the parallelepiped with edges OP, OQ, OR be V. Let V' be the volume of the parallelepiped which has O as one vertex and which has OP, OQ, OR as altitudes to three faces. Show that V V' = OP^{2}OQ^{2}OR^{2}. Generalize to n dimensions.

A3. All the complex numbers z_{n} are nonzero and z_{m}  z_{n} > 1 (for any m ≠ n). Show that ∑ 1/z_{n}^{3} converges.


A4. Take P inside the tetrahedron ABCD to minimize PA + PB + PC + PD. Show that ∠APB = ∠CPD and that the bisector of APB also bisects CPD.

A5. Let p(z) = z^{6} + 6z + 10. How many roots lie in each quadrant of the complex plane?


A6. Show that ∏_{1}^{∞} (1 + 2 cos(2z/3^{n})/3 = (sin z)/z for all complex z.


B1. Show that for any rational a/b ∈ (0, 1), we have a/b  1/√2 > 1/(4b^{2}).


B2. Do either (1) or (2)
(1) Prove that ∑_{2}^{∞} cos(ln ln n) / ln n diverges.
(2) Let k, a, b, c be real numbers such that a, k > 0 and b^{2} < ac. Show that ∫_{U} (k + ax^{2} + 2bxy + cy^{2})^{2} dx dy = π/( k √(ac  b^{2}) ), where U is the entire plane.


B3. C is a closed plane curve. If P, Q ∈ C, then PQ < 1. Show that we can find a disk radius 1/√3 which contains C.


B4. Let (1 + x  √(x^{2}  6x + 1) )/4 = ∑_{1}^{∞} a_{n}x^{n}. Show that all a_{n} are positive integers.


B5. a_{n} is a sequence of positive reals. Show that lim sup_{n→∞}( (a_{1} + a_{n+1})/a_{n})^{n} ≥ e.


B6. C is a closed convex curve. If P lies on C and T_{P} is the tangent at P, then T_{P} varies continuously with P. Let O be a point inside C. Given a point P on C, define f(P) to be the point where the perpendicular from O to T_{P} intersects C. Given P_{1}, define the sequence P_{n} by P_{n+1} = f(P_{n}). Assume that f is continuous and that, for each P, C lies entirely on one side of T_{P}. Show that P_{n} converges. Find S = { P : P = lim_{n→∞}P_{n} for some P_{1}}.
