USAMO 1992

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

Let $\mathrm{k}=1^{0}$. Show that $\Sigma_{0}^{88} 1 /(\cos \mathrm{nk} \cos (\mathrm{n}+1) \mathrm{k})=\cos \mathrm{k} / \sin ^{2} \mathrm{k}$.

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?

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

A complex polynomial has degree 1992 and distinct zeros. Show that we can find complex numbers $\mathrm{z}_{\mathrm{n}}$, such that if $\mathrm{p}_{1}(\mathrm{z})=\mathrm{z}-\mathrm{z}_{1}$ and $\mathrm{p}_{\mathrm{n}}(\mathrm{z})=\mathrm{p}_{\mathrm{n}-1}(\mathrm{z})^{2}-\mathrm{z}_{\mathrm{n}}$, then the polynomial divides $\mathrm{p}_{1992}(\mathrm{z})$.