# Difference between revisions of "2006 USAMO Problems"

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[[2006 USAMO Problems/Problem 4|Solution]] | [[2006 USAMO Problems/Problem 4|Solution]] | ||

=== Problem 2 === | === Problem 2 === | ||

− | A mathematical frog jumps along the number line. The frog starts at <math>1</math>, and jumps according to the following rule: if the frog is at integer <math>n</math>, then it can jump either to <math>n+1</math> or to <math>n+2^{m_n+1}</math> where <math>2^{m_n}</math> is the largest power of <math>2</math> that is a factor of <math>n</math>. Show that if <math>k | + | A mathematical frog jumps along the number line. The frog starts at <math>1</math>, and jumps according to the following rule: if the frog is at integer <math>n</math>, then it can jump either to <math>n+1</math> or to <math>n+2^{m_n+1}</math> where <math>2^{m_n}</math> is the largest power of <math>2</math> that is a factor of <math>n</math>. Show that if <math>k\ge 2</math> is a positive integer and <math>i</math> is a nonnegative integer, then the minimum number of jumps needed to reach <math>2^ik</math> is greater than the minimum number of jumps needed to reach <math> 2^i.</math> |

[[2006 USAMO Problems/Problem 5|Solution]] | [[2006 USAMO Problems/Problem 5|Solution]] |

## Revision as of 11:08, 12 July 2006

## Contents

## Day 1

### Problem 1

Let be a prime number and let be an integer with **.** Prove that there exists integers and with and

if and only if is not a divisor of **.**

Note: For a real number, let denote the greatest integer less than or equal to , and let denote the fractional part of x.

### Problem 2

For a given positive integer **k** find, in terms of **k**, the minimum value of for which there is a set of distinct positive integers that has sum greater than but every subset of size **k** has sum at most .

### Problem 3

For integral , let be the greatest prime divisor of . By convention, we set and . Find all polynomial with integer coefficients such that the sequence

is bounded above. (In particular, this requires for )

## Day 2

### Problem 1

Find all positive integers such that there are positive rational numbers satisfying

### Problem 2

A mathematical frog jumps along the number line. The frog starts at , and jumps according to the following rule: if the frog is at integer , then it can jump either to or to where is the largest power of that is a factor of . Show that if is a positive integer and is a nonnegative integer, then the minimum number of jumps needed to reach is greater than the minimum number of jumps needed to reach

### Problem 3

Let be a quadrilateral, and let and be points on sides and respectively, such that Ray meets rays and at and respectively. Prove that the circumcircles of triangles and pass through a common point.