Mock AIME 1 2006-2007 Problems/Problem 5

5. Let $p$ be a prime and $f(n)$ satisfy $0\le f(n) <p$ for all integers $n$ . $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$ . If for fixed $n$ , there exists an integer $0\le y < p$ such that:

$ny-p\left\lfloor \frac{ny}{p}\right\rfloor=1$

then $f(n)=y$ . If there is no such $y$ , then $f(n)=0$ . If $p=11$ , find the sum: $f(1)+f(2)+...+f(p^{2}-1)+f(p^{2})$ .

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Mock AIME 1 2006-2007 Problems/Problem 5