(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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