# Difference between revisions of "1992 USAMO Problems"

(Created page with '== Problem 1== Find, as a function of <math>\, n, \,</math> the sum of the digits of <cmath> 9 \times 99 \times 9999 \times \cdots \times \left( 10^{2^n} - 1 \right), </cmath> …') |
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<cmath> \frac{1}{\cos 0^\circ \cos 1^\circ} + \frac{1}{\cos 1^\circ \cos 2^\circ} + \cdots + \frac{1}{\cos 88^\circ \cos 89^\circ} = \frac{\cos 1^\circ}{\sin^2 1^\circ}. </cmath> | <cmath> \frac{1}{\cos 0^\circ \cos 1^\circ} + \frac{1}{\cos 1^\circ \cos 2^\circ} + \cdots + \frac{1}{\cos 88^\circ \cos 89^\circ} = \frac{\cos 1^\circ}{\sin^2 1^\circ}. </cmath> | ||

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[[1992 USAMO Problems/Problem 2 | Solution]] | [[1992 USAMO Problems/Problem 2 | Solution]] | ||

## Revision as of 10:51, 22 April 2010

## 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