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2006 USAMO Problems/Problem 1: Difference between revisions

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


if and only if '''<math>s</math>''' is not a divisor of '''<math>p-1</math>.'''
<center>
<math> \left\{ \frac{sm}{p} \right\} < \left\{ \frac{sn}{p} \right\} < \frac{s}{p} </math>
</center>
 
if and only if <math>\displaystyle s </math> is not a divisor of <math>\displaystyle p-1 </math>.
 
Note: For <math> \displaystyle 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 <math> \displaystyle x </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 ==
 
*[[2006 USAMO Problems]]
{{solution}}
 
== Resources ==
 
* [[2006 USAMO Problems]]
* [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=490569#p490569 Discussion on AoPS/MathLinks]

Revision as of 19:05, 1 September 2006

Problem

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

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

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

Note: For $\displaystyle x$ 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 $\displaystyle x$.

Solution

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Resources

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