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.

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

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

**Note:** \(\{x\}\) denotes the fractional part function.

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