# Questions tagged [floor-function]

The floor function, also known as the greatest integer function, maps a real number $x$ to the greatest integer less than or equal to $x$ (often denoted $\lfloor x \rfloor$). See also (ceiling-function).

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### Counting the number of solutions to an equation involving floor and ceiling function

I'd like to count the number of integer solutions to $x \lceil \mu y \rceil - y\lfloor \mu x \rfloor=K$, where $\lceil \cdot \rceil$ is the ceiling function, $\lfloor \cdot \rfloor$ is the floor ...
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### Equation involving floor function and fractional part function

How to solve $\frac{1}{\lfloor x \rfloor} + \frac{1}{\lfloor 2x \rfloor} = \{x\} + \frac{1}{3}$ , where $\lfloor \rfloor$ denotes floor function and {} denotes fractional part. I did couple of ...
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### Upper bound on $f(x)= \frac{\lfloor x\rfloor!}{\lfloor x+h\rfloor!} 2^{\lfloor x+h\rfloor!-\lfloor x\rfloor!}$ [closed]

I am trying to show that the following function is bounded: \begin{align} f(x)= \frac{\lfloor x\rfloor!}{\lfloor x+h\rfloor!} 2^{\lfloor x+h\rfloor!-\lfloor x\rfloor!} \end{align} for some $h>0$ ...
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### Evaluating $\int_0^{10\pi}(\lfloor \sec^{-1}x\rfloor + \lfloor \cot^{-1}x\rfloor)\,\mathrm{d}x$
My book has this problem: $$\int_0^{10\pi} (\lfloor \sec^{-1}x\rfloor + \lfloor \cot^{-1}x\rfloor) \;dx = \;?$$ Now, the domain of $\sec^{-1}x$ is $({-\infty}, -1) \bigcup (1, \infty)$. How can you ...
### $\lim_{n\to\infty}\frac{n -\big\lfloor\frac{n}{2}\big\rfloor+\big\lfloor\frac{n}{3}\big\rfloor-\dots}{n}$, a Brilliant problem
I encounter a question when visiting Brilliant: Find $\space\space\space\space\lim_{n\to\infty}s_n$ \$=\lim_{n\to\infty}\frac{n - \big \lfloor \frac{n}{2} \big \rfloor+ \big \lfloor \frac{n}{3} \big ...