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

 
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== Problem ==
 
== Problem ==
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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
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'''<center>{<math>\frac{sm}{p}</math>} < {<math>\frac{sn}{p}</math>}< <math>{\frac{s}{p}}</math></center>'''
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if and only if '''<math>s</math>''' is not a divisor of '''<math>p-1</math>.'''
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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 11:02, 12 July 2006

Problem

Let $p$ be a prime number and let $s$ be an integer with $0 < s < p$. Prove that there exists integers $m$ and $n$ with $0 < m < n < p$ and

{$\frac{sm}{p}$} < {$\frac{sn}{p}$}< ${\frac{s}{p}}$

if and only if $s$ is not a divisor of $p-1$.

Note: For $x$ a real number, let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$, and let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of x.

Solution

See Also

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