Greatest Integer Function

Okay, so it's pretty easy to see that the $x - [x]$ on the right hand side is nothing but the fractional part of $x$ , which I'll be writing as $\{ x\}$ . Using this knowledge and evaluating both sides gives:

$\frac { [2x][x] }{ [2x]+[x] } = \frac { 3{ x }-1 }{ 3 }$ .

Now, the fact that $0\le\{ x\}<1$ gives us something to work with, specifically, something with which we can figure out the bounds for $[x]$ ! But for that to work, it makes sense to convert $[2x]$ into an expression which is in terms of $[x]$ . We get:

$[2x]=2[x]\quad\quad\quad \quad for\quad \quad 0\le \{ x\} <0.5\\ [2x]=2[x]+1\quad \quad for\quad \quad 0.5\le \{ x\} <1$ .

And using this in the original re-evaluated equation gives:

$\frac { 3 }{ 2[x] }=\frac { 3\{ x\} -1 }{ 3 } \quad\quad\quad \quad\quad for\quad \quad 0\le\{ x\}<0.5\\ \\ \frac { 3[x]+1 }{ (2[x]+1)([x]) }=\frac { 3\{ x\} -1 }{ 3 } \quad \quad for\quad \quad 0.5\le\{ x\}<1$ .

Inputting the values of $\{x\}$ into the equations will give us some quadratic inequalities in $[x]$ , which can be easily solved. We can easily rule out the second case ( $0.5\le\{ x\}<1$ ), and consider the cases that will arise from the first (which can be solved to get $1.8<[x]<4.5$ ). These cases are $[x]=2, 3, 4$ . Substituting into the previous equation will give the respective values for $\{x\}$ , and with just a little bit of algebraic computation, the question is solved!

These are the values of $x$ that you should finally get:

$2\frac { 5 }{ 12 } ,3\frac { 1 }{ 6 } ,4\frac { 1 }{ 24 }$