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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.

# Problem Solving

How many $$0$$'s are on the end of $$180!?$$

Note: $$180!$$ is the factorial of $$180,$$ which means $$180\times179\times178\times\cdots\times2\times1.$$

The positive root of $20x^2=\lceil x \rceil$ is $$A.$$ Find the value of $$\frac{1}{A^2}.$$

Evaluate $\sum_{i=1}^{44}\left\lfloor\frac{16i}{45}\right\rfloor.$

Evaluate

$\sum_{i=1}^{ 128 } \lfloor \log_2 i \rfloor .$

Which of the figures above correctly illustrates the graph of $$y=\{x\}?$$

Note: $$\{x\}$$ denotes the fractional part function.

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