Floor Ceiling Ceiling Floor: JEE Style

This is the shortest way to solve the problem:

Plugging in integer values for $x$ quickly tells us that $29<x<30$ .

Assuming $x$ is not an integer: $\left\lceil x \right\rceil =\left\lfloor x \right\rfloor +1$

$\Rightarrow \left\lfloor 29x \right\rfloor +\left\lfloor 30x \right\rfloor =1728$

Now, using inequalities and properties of floor function, we have:

$1728\le 59\alpha <1730$

$1728<59\beta <1730$

$\Rightarrow 1728<30\alpha +29\beta <1730$

Since the answer is an integer, $30\alpha +29\beta$ is also an integer. Therefore, $30\alpha +29\beta+1=1729+1=1730$ .

For a more straightforward and rigorous approach, look at the solution to PI Han Goh's original problem .