Algebra
# Floor and Ceiling Functions

What is $\left\lfloor\sqrt{46}\right\rfloor?$

Which of the following statements is/are true?

I. $\lfloor x+n\rfloor=\lfloor x\rfloor+n$ for any real $x$ and any integer $n.$

II. $\lfloor x\rfloor+\lfloor-x\rfloor=0$ if and only if $x$ is an integer.

III. $\lfloor x\rfloor+\lfloor y\rfloor\le\lfloor x+y\rfloor\le\lfloor x\rfloor+\lfloor y\rfloor+1$ for any real $x$ and $y.$

What is the solution set to $\left\lfloor \frac{x}{2}\right\rfloor=4?$

How many integers $x$ satisfy the equation $\lfloor\sqrt{x}\rfloor=3?$

What is $\lfloor \pi \rfloor?$