Greatest Integer Function

Algebra Level 5

\(x\) is a real number and satisfies the equation:
\( \frac{1}{[x]}+\frac{1}{[2x]}=x-[x]+\frac{1}{3}\).
The sum of all such numbers can be expressed as \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m+n\)

Details and Assumptions

\([x]\) denotes the Greatest Integer Function: \([x]=n\), where \(n\) is the greatest integer less than \(x\).


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