Greatest Integer Function

Algebra Level 5

xx is a real number and satisfies the equation:
1[x]+1[2x]=x[x]+13 \frac{1}{[x]}+\frac{1}{[2x]}=x-[x]+\frac{1}{3}.
The sum of all such numbers can be expressed as mn\frac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n

Details and Assumptions

[x][x] denotes the Greatest Integer Function: [x]=n[x]=n, where nn is the greatest integer less than xx.

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