# (NIMO 2014) Reciprocal of Interval

Number Theory Level pending

For any interval $$\mathcal{A}$$ in the real number line not containing zero, define its $$\textit{reciprocal}$$ to be the set of numbers of the form $$\frac1x$$ where $$x$$ is an element in $$\mathcal{A}$$. Compute the number of ordered pairs of positive integers $$(m,n)$$ with $$m<n$$ such that the length of the interval $$[m,n]$$ is $$10^{10}$$ times the length of its reciprocal.

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