Easy floor's problem

We have

$\begin{aligned} & x + \left\lfloor 2x \right\rfloor = 3.14 \\ \implies & \left\lfloor 2x \right\rfloor = 3.14 - x \end{aligned}$

Note that the LHS of the above equation is an integer, therefore RHS must be an integer too. Thus we conclude that $\{x\} = 0.14$ and we can write $x = n + 0.14$ , where $n$ is an integer. Now we have

$\begin{aligned} & \left\lfloor 2n + 0.28 \right\rfloor = 3 - n \\ \implies & 2n + \left\lfloor 0.28 \right\rfloor = 3 - n \\ \implies & 2n = 3 - n \\ \implies & n = 1 \end{aligned}$

Thus we get $x = 1 + 0.14 = 1.14$ and so $\left\lfloor 1000x^2 \right\rfloor = \left\lfloor 1000 \times 1.2996 \right\rfloor = \boxed{1299}$ .

$For ~~1\geq~x~<2, \left\lfloor 2x \right\rfloor=2.\\ So~~x=3.14-2=1.14, ~~which ~IS~~ ~1\geq~x~<2.\\ \therefore~~ \left\lfloor1000*1.14^2 \right\rfloor =1299$

You can, rather easily, observe that the equation holds true for : $x=1.14$
So : $1000x^{2}=1299.6\Rightarrow \boxed{\lfloor 1000x^{2}\rfloor = 1299}$