# Difference between revisions of "1999 AIME Problems"

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== Problem 3 == | == Problem 3 == | ||

+ | Find the sum of all positive integers <math>\displaystyle n</math> for which <math>\displaystyle n^2-19n+99</math> is a perfect square. | ||

[[1999 AIME Problems/Problem 3|Solution]] | [[1999 AIME Problems/Problem 3|Solution]] |

## Revision as of 01:43, 22 January 2007

## Contents

## Problem 1

Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms also being prime.

## Problem 2

Consider the parallelogram with vertices and A line through the origin cuts this figure into two congruent polygons. The slope of the line is where and are relatively prime positive integers. Find

## Problem 3

Find the sum of all positive integers for which is a perfect square.

## Problem 4

## Problem 5

## Problem 6

## Problem 7

There is a set of 1000 switches, each of which has four positions, called , and . When the position of any switch changes, it is only from to , from to , from to , or from to . Initially each switch is in position . The switches are labeled with the 1000 different integers , where , and take on the values . At step i of a 1000-step process, the -th switch is advanced one step, and so are all the other switches whose labels divide the label on the -th switch. After step 1000 has been completed, how many switches will be in position ?