Floor and Ceiling Functions

Floor and Ceiling Functions: Level 1 Challenges


limxx2x= ?\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!60!.

For all x0x \ge 0, and xRx \in \mathbb {\text{R}}, is it true that

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

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

So, Find the sum of all possible solutions of

x+2x+4x+8x+16x+32x=12345\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 00

If f(x)=x2f(x) = \lceil x^{2}\rceil, what is the value of f(5)f\left(\sqrt {\sqrt {5}} \right)?

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


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