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Floor and Ceiling Functions

Floor functions map a real number x to the largest integer less than or equal to x. You can probably guess what the ceiling function does.

Level 1

\[\large \lim_{x\to \infty}\dfrac{\lfloor \sqrt x \rfloor^2}{x} = \ ? \]

Note: Type -1000 as an answer if you think this limit does not exist.

Find the number of trailing zeros of the number \(60!\).

For all \(x \ge 0\), and \(x \in \mathbb {\text{R}}\), is it true that

\[ \left \lfloor \sqrt{\lfloor x \rfloor}\right \rfloor = \left \lfloor \sqrt{x} \right \rfloor \]

Let \(\left\lfloor x \right\rfloor \) denote the greatest integer less that or equal to \(x\).

So, Find the sum of all possible solutions of

\[\left\lfloor x \right\rfloor+ \left\lfloor 2x \right\rfloor+\left\lfloor 4x \right\rfloor+\left\lfloor 8x \right\rfloor+\left\lfloor 16x \right\rfloor+\left\lfloor 32x \right\rfloor=12345\]

Note: If you feel that there are no such possible values of x, type the answer as \(0\)

If \(f(x) = \lceil x^{2}\rceil\), what is the value of \(f\left(\sqrt {\sqrt {5}} \right)\)? \[ \]

Notation: \( \lceil \cdot \rceil \) denotes the ceiling function.

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