# Difference between revisions of "2006 USAMO Problems"

Ragnarok23 (talk | contribs) |
Ragnarok23 (talk | contribs) |
||

Line 7: | Line 7: | ||

if and only if '''<math>s</math>''' is not a divisor of '''<math>p-1</math>.''' | if and only if '''<math>s</math>''' is not a divisor of '''<math>p-1</math>.''' | ||

− | Note: For <math>x</math> a real number, let <math>\lfloor x \rfloor</math> denote the greatest integer less than or equal to <math>x</math>, and let <math>\{x\} = x - \lfloor x \rfloor</math> denote the fractional part of x | + | Note: For <math>x</math> a real number, let <math>\lfloor x \rfloor</math> denote the greatest integer less than or equal to <math>x</math>, and let <math>\{x\} = x - \lfloor x \rfloor</math> denote the fractional part of x. |

+ | |||

+ | [[2006 USAMO Problem 1|Solution]] | ||

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

+ | For a given positive integer '''k''' find, in terms of '''k''', the minimum value of <math>N</math> for which there is a set of <math>2k+1</math> distinct positive integers that has sum greater than <math>N</math> but every subset of size '''k''' has sum at most <math>\frac{N}{2}</math>. | ||

+ | |||

+ | [[2006 USAMO Problem 2|Solution]] | ||

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

+ | For integral <math>m</math>, let <math>p(m)</math> be the greatest prime divisor of <math>m</math>. By convention, we set <math>p(\pm 1)=1</math> and <math>p(0)=\infty</math>. Find all polynomial <math>f</math> with integer coefficients such that the sequence | ||

+ | |||

+ | <math>(p(f(n^2))-2n)_{n\ge 0}</math> | ||

+ | |||

+ | is bounded above. (In particular, this requires <math>f(n^2)\neq 0</math> for <math>n\ge 0</math> |

## Revision as of 19:31, 4 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