Used for questions and equations involving the floor function, which is defined to be the function that returns the largest integer less than or equal to $x$ (often denoted by $\lfloor x\rfloor$). See also (ceiling-function).

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20 views

Floor function algebraic manipulation

Well, I'm working with a matrix $D=\vert d(j,x):1\leq j\leq m,1\leq x\leq 2^{p}\vert$ where $m=\frac{n}{2}$ and $j,x,p,n$ are positive integers and $n \equiv 2 \pmod 4$. Its elements are defined by: $...
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1answer
84 views

Every positive integer is a limit point of the sequence $a_n=n-\sqrt{n}\left\lfloor{\sqrt{n}}\right\rfloor$

I have the following sequence and limit: $$\left(a_n\right)=\left(n-\sqrt{n}\left\lfloor{\sqrt{n}}\right\rfloor\right)$$ $$\lim_{n\to\infty}{n\left(\sqrt{n^2+2\ell}-n\right)}=\ell,$$ where $\ell$ is ...
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1answer
32 views

How to simplify a sum involving the floor function

Let us suppose the positive integers $a$, $b$ and $n$ with $a<b$. Is it possible to simplify the following sum: $$2 \left\lfloor \frac{an}{b} \right\rfloor b + \frac{2^2}{3} \sum_{j=1}^{n-2} 3^j \...
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1answer
33 views

Value of a product of cosines and the floor of its reciprocal

$$ \mbox{The question states}\quad {a \over b} =\prod_{n = 1 \atop{\vphantom{\LARGE A}n \not= 9}}^{17}\cos\left(n\pi \over 18\right) $$ $$\mbox{And it is also provided that}\quad \left\lfloor{b \over ...
2
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2answers
86 views

About a proof that $\lfloor x^2\rfloor = \lfloor x\rfloor^2$ for unbounded non integer values of $x$

I am taking a first course in discrete mathematics. The instructor parsed the following question that has the following solution, respectively: Prove the statement: For all positive integers $N$, ...
3
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1answer
53 views

How to prove that $\lfloor \frac{n}{2}\rfloor$ = $\lceil \frac{n-1}{2}\rceil$

I'm having a hard time proving that: $$\left \lfloor \frac{n}{2}\right\rfloor = \left\lceil \frac{n-1}{2}\right\rceil$$ I've tried various algebraic manipulations. I've also tried to see if I could ...
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2answers
36 views

Solving an equation involving the root of a floor function of x

$x- ⌊\sqrt{x} ⌋^2 =10$. Prove that $x=35$ where $⌊x⌋$= floor of x
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0answers
40 views

Limit of an integral involving the floor function

I am trying to obtain an asymptotic expansion of $$\int_t^\infty \lfloor x\rfloor \frac {x}{\sqrt{x^2-t^2}} \ \Bbb d x$$ for $t \to \infty$ and $\lfloor x \rfloor $ denoting the floor function. I ...
6
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1answer
102 views

$\lfloor x\rfloor \cdot \lfloor x^2\rfloor = \lfloor x^3\rfloor$ means that $x$ is close to an integer

Suppose $x>30$ is a number satisfying $\lfloor x\rfloor \cdot \lfloor x^2\rfloor = \lfloor x^3\rfloor$. Prove that $\{x\}<\frac{1}{2700}$, where $\{x\}$ is the fractional part of $x$. My ...
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2answers
215 views

Solve in integers $x,y$ the equation $\left\lfloor\frac{xy-xy^2}{x+y^2} \right\rfloor=a$

Solve in integers the equation $$\left\lfloor\frac{xy-xy^2}{x+y^2} \right\rfloor=a$$ My work so far: 1) If $a=1$, then $x\in\{-1;-2;-3\}$. i) $x=-1 \Rightarrow y\ge-3$ ii) $x=-2 \Rightarrow y\...
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2answers
31 views

The greatest value of $\cos(xe^{\lfloor x \rfloor} +7x^2 -3x)$

The greatest value of $g(x)=\cos(xe^{\lfloor x \rfloor} +7x^2 -3x)$ , $x\in [-1,\infty)$ , is My work: For function to be maximum $f(x) = xe^{\lfloor x \rfloor} +7x^2 -3x$ must be minimum When $ x ...
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2answers
101 views

Find the integer part of the sum $S=\sum_{k=1}^{80} \frac{1}{\sqrt k} $

Let $$S=\sum_{k=1}^{80} \frac{1}{\sqrt k}.$$Then I would like to obtain $\lfloor S \rfloor$, the integer part of $S$. I am not able to think how to start question .
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1answer
96 views

Necessary and sufficient condition that $ \lceil \sqrt { \lfloor x \rfloor } \rceil = \lceil \sqrt { x } \rceil $

I am stumped at the following paragraph, which comes from Concrete Mathematics, Chapter 3, Section 2, Page 73: What is a necessary and sufficient condition that $ \lceil \sqrt { \lfloor x \rfloor }...
4
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2answers
44 views

Find the limit: $\lim_{n\to\infty} \frac{ \sum_{i=1}^n\lfloor i^3x \rfloor}{n^4}$

$$\lim_{n \to {\infty}} \frac{ \sum_{i=1}^n\lfloor i^3x \rfloor}{ n^4}$$ My work $$\lim_{n \to {\infty}} \frac{ \sum_{i=1}^ni^3x}{ n^4} -\lim_{n \to {\infty}} \frac{ \sum_{i=1}^n{\{i^3x\}}}{ n^4}$$ $$...
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2answers
54 views

Prove that $[\sqrt{n}+\sqrt{n+1}]=[\sqrt{4n+1}]$

For every real positive n prove that $\sqrt{4n+1}<\sqrt{n}+\sqrt{n+1}<\sqrt{4n+2}$.Hence,or otherwise prove that $[\sqrt{n}+\sqrt{n+1}]=[\sqrt{4n+1}]$. Wher [x] denotes the greatest integer not ...
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1answer
29 views

Evaluate the given limit

$$f(x):=\begin{cases}\frac{ \sin \lfloor x+1\rfloor }{ \lfloor x+1\rfloor } & \lfloor x+1\rfloor \ne0 \\ 0 & \lfloor x+1\rfloor=0 \ \end{cases}$$ Then at $x=-1$ find the ...
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4answers
71 views

Find the number of solutions to $ \lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 4x \rfloor + \cdots + \lfloor 32x \rfloor =12345$

Find the number of solutions of the equation $$ \lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 8x \rfloor + \lfloor 16x \rfloor + \lfloor 32x \rfloor =12345$$ Here, $ \...
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1answer
54 views

How to integrate the following floor function?

I need to calculate the following integral, $$\int_{1}^{10}\frac{x-\lfloor x \rfloor }{x^2} dx$$. So I did this- $$\int_{1}^{10}\frac{x-\lfloor x \rfloor }{x^2} dx=\int_{1}^{10}\frac{1 }{x} dx-\int_{...
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4answers
175 views

Evaluate $\int_{0}^{\infty} (-1)^{\lfloor x\rfloor}\cdot e^{-x} dx $ [closed]

I'm having trouble integrating the following: $$\int_{0}^{\infty} (-1)^{\lfloor x \rfloor}\cdot e^{-x} \, \mathrm{d}x $$ where $\lfloor x \rfloor$ denotes the floor of $x$. Can you help please?
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0answers
76 views

Find all integers $p>q\ge 0$, such that for all real $x,y \in [0;1]$ following inequality holds $[px]+[py]\ge [qx+y]+[x+qy]$

Find all integers $p>q\ge 0$, such that for all real $x,y \in [0;1]$ following inequality holds $$\lfloor px \rfloor + \lfloor py\rfloor \ge \lfloor qx+y\rfloor+\lfloor x+qy \rfloor$$ I used $x\...
0
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1answer
25 views

Analyzing and plotting function $f(x)=\lfloor\frac{3-x}{1+x^2}\rfloor$?

How to plot and analyze function $f(x)=\lfloor\frac{3-x}{1+x^2}\rfloor$? How to prove $\lim_{x\to -\infty}f(x)=0$ and $\lim_{x\to \infty}f(x)=-1$? My work: 1) Function is well defined for every $x\...
3
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1answer
95 views

What is $\sum_{i=0}^n \left\lfloor \sqrt{i}\right\rfloor \binom{n}{i}$?

Since both $\sum_{i=0}^n \left\lfloor \sqrt{i}\right\rfloor$ and $\sum_{i=0}^n \binom{n}{i}$ have simple closed-form evaluations, it is natural to consider the evaluation of the binomial sum $\sum_{...
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0answers
31 views

Probability of collision of some family of hash functions

Given $x$ and $y$ in $\mathbb{R}$, and let $\mathcal{H} = \{ h \mid \mathbb{R} \to \mathbb{N} \}$ be a family of hash functions where $ h(x) = \left\lfloor x + \sum^C_{i=1} U_i \right\rfloor$ for some ...
2
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2answers
57 views

Value of $z$ in the given system of equations

If $$\{x\}+y+\lfloor{z}\rfloor=3.1$$ $$x+\lfloor{y}\rfloor+\{z\}=2.4$$ $$\lfloor{x}\rfloor+\{y\}+z=1.3$$ then find the value of $z$. My attempt: I converted fractional part of every equation to ...
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1answer
173 views

A form for a piecewise continuous function?

$\def\rr{\mathbb{R}}$Take any $D \subseteq \rr$. Is it true that for any piecewise continuous function $f : D \to \rr$ there is an infinitely differentiable continuous function $g : D \to \rr$ and a ...
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2answers
77 views

Solve in integers the equation $\left\lfloor\frac{x^2-y^3}{x+y^2} \right\rfloor=1+x-y$

Solve in integers the equation $$\left\lfloor\frac{x^2-y^3}{x+y^2} \right\rfloor=1+x-y$$ My attempt: I used http://www.wolframalpha.com/: $x=-2; y=\{3,4,5,6,7,..\}$ or $x=-1, y=\{-10,-9,...\}$. ...
2
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1answer
33 views

Evaluation of $\bigg[ \lim_{x\to 0} \frac{e-(1+x)^{1/x}}{\tan x}\bigg]$

Find the value of $$\bigg[ \lim_{x\to 0} \frac{e-(1+x)^{1/x}}{\tan x}\bigg]$$ where $[.]$ represents floor function (or greatest integer function) I wrote it as $\bigg[ \lim_{x\to 0} \frac{e-(1+x)^...
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3answers
71 views

How can I remove the floor function from ⌊ab/10⌋?

After several weeks of trying on my own, I was hoping for a hand. I am familiar with transitioning ⌊ab/10⌋ to a mod function as well as a trig function. Ideally, I would like a solution that involved ...
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1answer
40 views

Removing dicontinuity from functions involving modulo?

I am currently looking into removing discontinuity from piecewise continuous functions without changing the derivative where it is defined and (preferably) the value of right sided limit at 0. This is ...
0
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1answer
89 views

Removing jump discontinuity from a tricky function.

I have the function $\cos(x)\lfloor x \rfloor$ which I would like to make continuous without changing the derivative where it exists or the values approaching 0 from the right side. I can do this by ...
0
votes
1answer
90 views

Example of a jump discontinuity where the left and right hand limits do not exist? [closed]

Right off the bat I should probably mention that I am speaking more visually rather than in manners that can be proven rigorously. Please keep that in mind when reading. I'm looking for a function ...
2
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1answer
70 views

Is the limit $\lim_{x\rightarrow0}\frac{\sin{[x]}}{[x]}$ a one sided limit or not?

Is the limit $\displaystyle\lim_{x\rightarrow0}\frac{\sin{[x]}}{[x]}$ a one sided limit or not? Here $[\, \cdot\, ]$ is the greatest integer function. According to me the right hand limit will be not ...
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1answer
77 views

Property of the floor function

Given $u,v \in \mathbb{R}_{+}$, and let $n:=\lfloor v \rfloor$. where $u\in [0,1]$. Is $\lfloor vu \rfloor =\lfloor nu \rfloor$ or $\lfloor vu \rfloor =\lfloor nu \rfloor+1$? Edit: I missed the ...
1
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2answers
26 views

Limit of floor function

I would like to calculate these limits $\lim\limits_{x \to 0^{+}} \frac{x}{a}{\big\lfloor\frac{b}{x}\big\rfloor}$ $\lim\limits_{x \to 0^{+}} \frac{b}{x}{\big\lfloor\frac{x}{a}\big\rfloor}$, where $a,...
0
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2answers
67 views

Solutions to $\lfloor 2x\rfloor-\lfloor x+1\rfloor=2x$

Find all solutions to $$[2x]-[x+1]=2x$$ where $[x]=\lfloor x\rfloor$ $$$$ I divided this into 2 cases: $$Case 1:x=[x]+\{x\}\text{ where } 0\le\{x\}<0.5$$ $$Case 1:x=[x]+\{x\}\text{ where } 0.5\...
4
votes
2answers
127 views

Solutions to $\dfrac{1}{\lfloor x\rfloor}+\dfrac{1}{\lfloor 2x\rfloor}=\{x\}+\dfrac{1}{3}$

Find all solutions to $$\dfrac{1}{\lfloor x\rfloor}+\dfrac{1}{\lfloor 2x\rfloor}=\{x\}+\dfrac{1}{3}$$ $$$$ Unfortunately I have no idea as to how to go about this. On rearranging, I got $$3\lfloor ...
3
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2answers
79 views

Solution of $x-1=(x-\lfloor x \rfloor)(x-\{x\})$

Find all solutions for $$x-1=(x-\lfloor x \rfloor)(x-\{x\})$$ $$$$My approach: $$x-1=\lfloor x \rfloor\{x\}$$ $$\dfrac{x-1}{\lfloor x \rfloor}=\{x\}$$ $$\Rightarrow 0\le \dfrac{x-1}{\lfloor x \...
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1answer
46 views

Series expansion of {x}

Hello and sorry for my bad English. I am not mathematician, so sorry if this seems a silly question. I've seen this formula regarding the fractional part of a number in Wikipedia, and I would like to ...
3
votes
2answers
110 views

Solutions to $\lfloor x\rfloor\lfloor y\rfloor=x+y$

Find all solutions to $$\lfloor x\rfloor\lfloor y\rfloor=x+y$$ and show that the non-Integral solutions lie on two unique lines. Also determine the equations of these 2 lines. I divided the problem ...
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1answer
29 views

Summation Closed form for floor$\left(\log_n\right)$

The closed sum for the floors of logs of consecutive integers is: $$\sum_{i=0}^n \lfloor \log_2i\rfloor = n\lfloor \log_2n\rfloor-2^{\lfloor \log_2n\rfloor+1}+\lfloor \log_2n\rfloor+2$$ This formula ...
0
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2answers
54 views

Integration of floor function.

$\displaystyle{\int_{0}^{3\pi}\left\lfloor\sin\left(x\right)\right\rfloor\,\mathrm{d}x = -\pi \approx -3.14159}$ Here, $\left\lfloor\cdots\right\rfloor$ means floor function Why is it so ?. ...
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1answer
39 views

Prove $\Sigma_{k=0}^{n-1}\lfloor x+\frac{k}{n}\rfloor=\lfloor nx\rfloor$ , n is a Natural Number

Prove the following identity: $$\lfloor x\rfloor +\lfloor x+\frac{1}{n}\rfloor +\lfloor x+\frac{2}{n}\rfloor +\lfloor x+\frac{3}{n}\rfloor+...+\lfloor x+\frac{n-1}{n}\rfloor =\lfloor nx\rfloor$$ ...
1
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3answers
106 views

Solutions of $\lfloor 4x\rfloor+\lfloor 3x\rfloor=1$

Find all solutions of $$\lfloor 4x\rfloor+\lfloor 3x\rfloor=1$$ I have no idea as to how to go about this question. I would be grateful if somebody would please show me how to solve such questions. ...
3
votes
1answer
58 views

A little help with $ x_n = \frac{1}{n^2} \sum_{k=1}^{n} \left[ k \alpha \right] $

So I have this little problem I want to solve that says the following: For every number $\alpha \in \mathbb{R} $, analyze the following sequence $$ \{ x_n \} = \frac{1}{n^2} \sum_{k=1}^{n} \left[ k \...
2
votes
2answers
99 views

Analytic floor function, why this seems to work?

I have been using this formula which I determined for myself for quite some time now for use in everything from the sgn() function to the Kronecker delta to the ceiling and NINT() functions but haven'...
1
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2answers
34 views

Difference of the floor of a product and the product of floors

Is there any way the following can be simplified? $$\lfloor f(x)\cdot g(x) \rfloor - \lfloor f(x) \rfloor \cdot \lfloor g(x) \rfloor$$
1
vote
2answers
86 views

Integrating the floor of a function

The integral that I'm trying to simplify is this: (both $x$ and $c$ are natural numbers, if that helps) $$ \mathrm{F}\left(x,c\right) \equiv \int_{0}^{c}\left\lbrace\vphantom{\LARGE A}% \left\lfloor ...
1
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0answers
36 views

Does $g$ map $\mathbb{R}$ onto the Cantor set?

For $x\in\mathbb{R}$ define \begin{equation} g(x)=1+\tfrac{3}{2}\sum_{k=0}^{\infty}\left(\frac{\left\lfloor2^{2k}x\right\rfloor}{2^{2k}}-\frac{\left\lfloor2^{2k+1}x\right\rfloor}{2^{2k+1}}\right) \...
-1
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1answer
45 views

How does this faulty system of integration change the nature of jump discontinuity?

Let's define a sort of faulty integral. For the purposes of this question we shall assume that this is the regular integral. This integral integrates all functions properly however it's gets confused ...
0
votes
1answer
21 views

Removing Discontinuity in 3-space without changing the partial derivative

Is it possible to find a version of the function $$f(x,y) = x\cdot \lfloor y \rfloor + \lfloor x\rfloor^2$$ That is continuous. ANY operation is allowed in changing the function as long as the ...