# 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 $$x$$.

×