# 2006 USAMO Problems

## 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.