# Difference between revisions of "2006 USAMO Problems/Problem 1"

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== Problem == | == Problem == | ||

+ | Let '''<math>p</math>''' be a prime number and let '''<math>s</math>''' be an integer with '''<math>0 < s < p</math>.''' Prove that there exists integers '''<math>m</math>''' and '''<math>n</math>''' with '''<math>0 < m < n < p</math>''' and | ||

+ | |||

+ | '''<center>{<math>\frac{sm}{p}</math>} < {<math>\frac{sn}{p}</math>}< <math>{\frac{s}{p}}</math></center>''' | ||

+ | |||

+ | 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. | ||

== Solution == | == Solution == | ||

== See Also == | == See Also == | ||

*[[2006 USAMO Problems]] | *[[2006 USAMO Problems]] |

## Revision as of 12:02, 12 July 2006

## Problem

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.