2006 USAMO Problems

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Day 1

Problem 1

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

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