

A1. A cone has circular base radius 1, and vertex a height 3 directly above the center of the circle. A cube has four vertices in the base and four on the sloping sides. What length is a side of the cube?


A2. Let C be the circle center (0, 0), radius 1. Let X, Y be two points on C with positive x and y coordinates. Let X_{1}, Y_{1} be the points on the xaxis with the same xcoordinates as X and Y respectively, and let X_{2}, Y_{2} be the points on the yaxis with the same ycoordinates. Show that the area of the region XYY_{1}X_{1} plus the area of the region XYY_{2}X_{2} depends only on the length of the arc XY, and not on its position.


A3. Let R be the reals. Let f : R → R have a continuous third derivative. Show that there is a point a with f(a) f '(a) f ''(a) f '''(a) ≥ 0.


A4. Define the sequence of decimal integers a_{n} as follows: a_{1} = 0; a_{2} = 1; a_{n+2} is obtained by writing the digits of a_{n+1} immediately followed by those of a_{n}. When is a_{n} a multiple of 11?


A5. A finite collection of disks covers a subset X of the plane. Show that we can find a pairwise disjoint subcollection S, such that X ⊆ ∪{3D : D ∈ S}, where 3D denotes the disk with the same center as D and 3 times the radius.


A6. P, Q, R are three (distinct) lattice points in the plane. Prove that if (PQ + QR)^{2} < 8 area PQR + 1, then P, Q, R are vertices of a square.


B1. Find the minimum of { (x + 1/x)^{6}  (x^{6} + 1/x^{6})  2 }/{ (x + 1/x)^{3} + (x^{3} + 1/x^{3}) }, for x > 0.


B2. Let P be the point (a, b) with 0 < b < a. Find Q on the xaxis and R on y = x, so that PQ + QR + RP is as small as possible.


B3. Let S be the sphere center the origin and radius 1. Let P be a regular pentagon in the plane z = 0 with vertices on S. Find the surface area of the part of the sphere which lies above (z > 0) P or its interior.


B4. For what m, n > 0 is ∑_{0}^{mn1} (1)^{[i/m] + [i/n]} = 0?


B5. Let n be the decimal integer 11...1 (with 1998 digits). What is the 1000th digit after the decimal point of √n?


B6. Show that for any integers a, b, c we can find a positive integer n such that n^{3} + a n^{2} + b n + c is not a perfect square.

