Floor and Ceiling Functions

Floor Function


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

Which of the following statements is/are true?

I. x+n=x+n\lfloor x+n\rfloor=\lfloor x\rfloor+n for any real xx and any integer n.n.
II. x+x=0\lfloor x\rfloor+\lfloor-x\rfloor=0 if and only if xx is an integer.
III. x+yx+yx+y+1\lfloor x\rfloor+\lfloor y\rfloor\le\lfloor x+y\rfloor\le\lfloor x\rfloor+\lfloor y\rfloor+1 for any real xx and y.y.

What is the solution set to x2=4?\left\lfloor \frac{x}{2}\right\rfloor=4?

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

What is π?\lfloor \pi \rfloor?


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