

1. Let a_{n} be the number written with 2^{n} nines. For example, a_{0} = 9, a_{1} = 99, a_{2} = 9999. Let b_{n} = ∏_{0}^{n} a_{i}. Find the sum of the digits of b_{n}.


2. Let k = 1^{o}. Show that ∑_{0}^{88} 1/(cos nk cos(n+1)k ) = cos k/sin^{2}k.


3. A set of 11 distinct positive integers has the property that we can find a subset with sum n for any n between 1 and 1500 inclusive. What is the smallest possible value for the second largest element?


4. Three chords of a sphere are meet at a point X inside the sphere but are not coplanar. A sphere through an endpoint of each chord and X touches the sphere through the other endpoints and X. Show that the chords have equal length.


5. A complex polynomial has degree 1992 and distinct zeros. Show that we can find complex numbers z_{n}, such that if p_{1}(z) = z  z_{1} and p_{n}(z) = p_{n1}(z)^{2}  z_{n}, then the polynomial divides p_{1992}(z).

