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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.

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

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

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

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

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

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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 \(0\)

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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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