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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 4

         

\[ \large \sum_{k=1}^{202} \left \lceil \sqrt k \ \right \rceil = \ ? \]

Find the positive integer \(n\), for which \[\lfloor \log_2{1}\rfloor+\lfloor\log_2{2}\rfloor+\lfloor\log_2{3}\rfloor+\cdots+\lfloor\log_2{n}\rfloor=1994.\]

What is the minimum integer value of \(x\) that satisfies the equation

\[\lfloor{\sqrt{x}}\rfloor - \lfloor{\sqrt{x+34}}\rfloor=0 ? \]

This problem is posed by Siam H.

Details and assumptions

The function \(\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z}\) refers to the greatest integer smaller than or equal to \(x\). For example \(\lfloor 2.3 \rfloor = 2\) and \(\lfloor -5 \rfloor = -5\).

\[\left\lfloor x+0.19 \right\rfloor +\left\lfloor x+0.20 \right\rfloor +\left\lfloor x+0.21 \right\rfloor + \ldots + \left\lfloor x+0.91 \right\rfloor =542 \]

If \(x\) satisfies the equation above, find the value of \(\left\lfloor 100x \right\rfloor \).

Note that \(\left\lfloor X \right\rfloor\) denote the floor function of \(X\).

The number of real solutions of \(7\lfloor x\rfloor + 23\{x\}=191\) is?

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