2006 USAMO Problems/Problem 1

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 $\displaystyle x$ .

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2006 USAMO Problems/Problem 1

Problem

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