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Number Theory Level 4

Find the number of real solutions of the equation 1 [ x ] + 1 [ 2 x ] = x − [ x ] + 1 3 . \frac{1}{[x]}+\frac{1}{[2x]}= x-[x]+\frac{1}{3}.

Note:
1. [ x ] [x] stands for the greatest integer function.
2. This problem is not original.


The answer is 3.

Sagnik Saha by Sagnik Saha , 23, India

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1 solution

Shrihari B
Dec 27, 2014

Take two cases Case 1: {x}<0.5
Case 2: {x}>0.5 Observe in Case 1 : [2x] =2[x]. Now solve. In case 2 : [2x]=2[x]+1

Now solve the LHS. You will get your three solutions. Cheers !!!

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