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

Revision as of 19:24, 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

This problem needs a solution. If you have a solution for it, please help us out by adding it.

Resources

@@ Line 19: / Line 19: @@
 * [[2006 USAMO Problems]]
 * [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=490569#p490569 Discussion on AoPS/MathLinks]
+[[Category:Olympiad Number Theory Problems]]

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

Revision as of 19:24, 1 September 2006

Problem

Solution

Resources