# 1992 USAMO Problems

## Problem 1

Find, as a function of the sum of the digits of where each factor has twice as many digits as the previous one.

## Problem 2

Prove

## Problem 3

For a nonempty set of integers, let be the sum of the elements of . Suppose that is a set of positive integers with and that, for each positive integer , there is a subset of for which . What is the smallest possible value of ?

## Problem 4

Chords , , and of a sphere meet at an interior point but are not contained in the same plane. The sphere through , , , and is tangent to the sphere through , , , and . Prove that .

## Problem 5

Let be a polynomial with complex coefficients which is of degree and has distinct zeros.Prove that there exists complex numbers such that divides the polynomial