# Difference between revisions of "2002 USAMO Problems"

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=== Problem 5 === | === Problem 5 === | ||

− | Let <math>a, b </math> be integers greater than 2. Prove that there exists a positive integer <math>k </math> and a finite sequence <math>n_1, n_2, \ldots, n_k </math> of positive integers such that <math>n_1 = a</math>, <math>n_k = b </math>, and <math> | + | Let <math>a, b </math> be integers greater than 2. Prove that there exists a positive integer <math>k </math> and a finite sequence <math>n_1, n_2, \ldots, n_k </math> of positive integers such that <math>n_1 = a</math>, <math>n_k = b </math>, and <math>n_in_{i+1} </math> is divisible by <math>n_i + n_{i+1} </math> for each <math>i </math> (<math> 1 \le i < k </math>). |

* [[2002 USAMO Problems/Problem 5 | Solution]] | * [[2002 USAMO Problems/Problem 5 | Solution]] | ||

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* [[USAMO Problems and Solutions]] | * [[USAMO Problems and Solutions]] | ||

* [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27&year=2002 2002 USAMO Problems on the Resources page] | * [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27&year=2002 2002 USAMO Problems on the Resources page] | ||

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## Latest revision as of 12:45, 31 March 2018

## Contents

## Day 1

### Problem 1

Let be a set with 2002 elements, and let be an integer with . Prove that it is possible to color every subset of either blue or red so that the following conditions hold:

(a) the union of any two red subsets is red;

(b) the union of any two blue subsets is blue;

(c) there are exactly red subsets.

### Problem 2

Let be a triangle such that

,

where and denote its semiperimeter and inradius, respectively. Prove that triangle is similar to a triangle whose side lengths are all positive integers with no common divisor and determine those integers.

### Problem 3

Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree with real coefficients is the average of two monic polynomials of degree with real roots.

## Day 2

### Problem 4

Let be the set of real numbers. Determine all functions such that

for all pairs of real numbers and .

### Problem 5

Let be integers greater than 2. Prove that there exists a positive integer and a finite sequence of positive integers such that , , and is divisible by for each ().

### Problem 6

I have an sheet of stamps, from which I've been asked to tear out blocks of three adjacent stamps in a single row or column. (I can only tear along the perforations separating adjacent stamps, and each block must come out of the sheet in one piece.) Let be the smallest number of blocks I can tear out and make it impossible to tear out any more blocks. Prove that there are real constants and such that

for all .

## Resources

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.