The function that maps a real number $x$ to the largest integer not greater than $x$. See also (ceiling-function).

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1answer
35 views

Find Limit floor (sin x) / floor(x) as x approaches 0.

I am unable to evaluate this limit. The floor function is giving me trouble. Any help will be appreciated. And please edit it so that it looks readable.
0
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2answers
32 views

Quick floor function

This isn't true, right? $$k\left\lfloor\frac n {2k}\right\rfloor\leq \left\lfloor\frac n k\right\rfloor$$ $2<k\leq \left\lfloor\dfrac {n-1} 2\right\rfloor$, $n>4$, $k,n$ are coprime.
0
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2answers
25 views

Floor Function Bound?

I am trying to prove or disprove the following bound: $2+\left(n-\left\lfloor\dfrac n k\right\rfloor k\right)\ge \left\lfloor\dfrac n k\right\rfloor$, where $2<k\le \left\lfloor\dfrac {n-1} ...
1
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1answer
40 views

Algebraic manipulation of floors and ceilings

I am trying to solve the summation $$ \sum_{n=3}^{\infty} \frac{1}{2^n} \sum_{i=3}^{n} \left\lceil\frac{i-2}{2}\right\rceil $$ I will list some of the simplifications that I've found so far, and ...
3
votes
1answer
59 views

Number of solutions to equation involving floor-function

For school I have to solve some problems involving floor-functions, and I have no clue how to solve this one: for a given $n$, calculate the number of $k$'s, $k\lt n$ such that the number of multiples ...
1
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1answer
48 views

A problem of sum floors

let $n$ be a positive integer, prove that $$\sum_{i=0}^{\left\lfloor\frac{n}{3}\right\rfloor}\left\lfloor\frac{n-3i}{2}\right\rfloor=\left\lfloor\frac{n^2+2n+4}{12}\right\rfloor.$$ It looks like we ...
0
votes
3answers
52 views

How to derive this inequality?

How to derive the following inequality for all positive integers $n \geq 2$? $$ \frac{n!}{n^n} \leq \left(\frac{1}{2}\right)^k,$$ where $k$ denotes the greatest integer less than or equal to $\dfrac ...
0
votes
1answer
50 views

limits involving greatest integer function

What can you say about the following limit: $$\lim_{x\rightarrow 0}\left(\left[\dfrac{100x}{\sin x}\right]+\left[\dfrac{99\sin x}{x}\right]\right)$$ where [.] represents the greatest integer function ...
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1answer
40 views

Comparing floor and ceiling fractions

Is the following true for all integers x>1: $\lfloor{\frac{2x}{3}}\rfloor \geq \lceil \frac{x}{2}\rceil$
4
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1answer
42 views

Prove that $\lceil \frac{\sqrt{n^2+1+\sqrt{n^2}}}{\sqrt{n^2+3+\sqrt{n^2+2}}-\sqrt{n^2+2+\sqrt{n^2+1}}}\rceil = 2n^2+n+3$

First, the question: Prove that $$\Bigg\lceil \frac{\sqrt{n^2+1+\sqrt{n^2}}}{\sqrt{n^2+3+\sqrt{n^2+2}}-\sqrt{n^2+2+\sqrt{n^2+1}}}\Bigg\rceil = 2n^2+n+3$$ The motive to this question is the ...
2
votes
1answer
36 views

Divisibility of $\lfloor (1+\sqrt{3})^{2m+1}\rfloor$ by $2^k$

I want to prove that the largest exponent $k$ of $2^k$ such that $2^k|\lfloor (1+\sqrt{3})^{2m+1}\rfloor$ is $m+1$ if $m\geq 1$. My instinct tells me that I should regard $(1+\sqrt{3})^{2m+1}$ as ...
1
vote
1answer
46 views

Proof for $\sum_{x=1}^{n-1}\lfloor \dfrac{mx}{n}\rfloor=\dfrac{(n-1)(m-1)}{2}$ where $(m,n)=1$

This identity might be well-known, but I could find the proof neither by myself not by searching it in Internet. Could you describe an outline of solution?
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votes
1answer
26 views

Expectation of a floor function applied to a continuous uniform random variable

Let $X$ be a random variable, uniformly distributed continuously in the interval $(x_0,x_1), x_0,x_1\in\mathbb{R}$ Obviously, its expectation $E[X]=0.5(x_0 + x_1)$. Apply round-to-nearest-integer to ...
0
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0answers
20 views

How to find the minima for $y = x^2 + a.x - \lfloor\sqrt{x^2+a.x - b}\rfloor^2$?

Please guide in how to find the value of $x$ for which $y = x^2 + a.x - \lfloor\sqrt{x^2+a.x - b}\rfloor^2$ will be minimum. I know this involves differentiation but am not sure on how to ...
3
votes
3answers
71 views

Values of $n$ for which $\lfloor 2 x\rfloor +\lfloor 4 x\rfloor +\lfloor 8 x\rfloor +\lfloor 20 x\rfloor =n$ has a solution

$$\lfloor 2 x\rfloor +\lfloor 4 x\rfloor +\lfloor 8 x\rfloor +\lfloor 20 x\rfloor =n$$ How would you find the values of $n$ for which the equation has a solution under the condition that $n \leq ...
4
votes
3answers
89 views

Evaluating $\lfloor (3 + \sqrt{5})^{34} \rfloor \pmod {100}$

The problem is to evaluate $\lfloor (3 + \sqrt{5})^{34} \rfloor \pmod {100}$ No calculators are allowed. I think I have to get rid of $\sqrt{5}$ somehow since it is irrational and would make it ...
2
votes
3answers
97 views

Summation with Ceilinged Logarithmic Function

According to Johann Blieberger's paper - "Discrete Loops and Worst Case Performance" (1994): $$ \sum_{i = 1}^{n}\left \lceil \log_2{(i)} \right \rceil = n\left \lceil \log_2{(n)} \right \rceil - ...
0
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0answers
19 views

Proving function defined by algorithm is convex

I'm working on my bachelor thesis and I'm trying to prove a conjecture, but I seem to miss the hint that helps me. I have an algorithm that defines a function $f:\mathbb{R}_{\geq ...
3
votes
1answer
42 views

Prove the sequences $\lfloor \alpha n\rfloor $ and $\lfloor \beta n\rfloor $ are disjoint

Here is another problem from a problem set that I can't solve. Let $\alpha$ and $\beta$ be irrational positive numbers such that $\frac{1}{\alpha}+\frac{1}{\beta}=1$ Prove that the sets $\{ ...
1
vote
1answer
48 views

Solve $\lfloor x^2 + 2x \rfloor = \lfloor x^2 \rfloor + 2 \lfloor x \rfloor$

How do we find all real $x$ such that $\lfloor x^2 + 2x \rfloor = \lfloor x^2 \rfloor + 2 \lfloor x \rfloor$, where $\lfloor \space \rfloor$ denotes the "greatest integer function" ?
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0answers
26 views

Discontinuity of floor function

I am still getting confused with showing discontinuity of functions, here is my attempt at a question , if someone could check to see if I am going about it in the correct way. $f: \mathbb{R} ...
1
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1answer
58 views

Finding the value of one-sided limits and greatest integer function.

$$ \lim_{x \to 0} \frac{a}{x} \left\lfloor\frac{x}{b} \right\rfloor $$ The $\lfloor \rfloor$ stands for the greatest integer function. I have calculated and the left-hand limit is coming as ...
0
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0answers
51 views

Definite integral of greatest integer function

I need to find the area under a function modeled by $f(x)=\left\lfloor 2.4x \right\rfloor+5$. I can't seem to figure out what the antiderivative of this is, so I'm going try to use a right Riemann ...
0
votes
2answers
22 views

Modeling a greatest integer function

I'm trying to model a function that resembles a greatest integer function. The domain is from [0, $\infty$). The inputs from 0 to 1.5 (non-inclusive) need to be mapped to an output of 0, and 1.5 to ...
1
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0answers
35 views

Full series expansion of the floor function

We know if $x$ is not an integer we have $$\left \lfloor x \right \rfloor=x-\frac{1}{2}+\frac{1}{\pi }\sum_{k=1}^{\infty}\frac{\sin(2\pi kx)}{k}$$ Is there an series expansion of floor function ...
2
votes
2answers
64 views

Values $nx - [nx]$ are distinct for an irrational number

When reading the proof for Dirichlet's Approximation Theorem, I came across the following statement: If $x$ is irrational, then $nx - [nx]$ are distinct for all $n \in \mathbb{Z}$. I don't ...
3
votes
0answers
44 views

Finding sum for function with floor-function

I am trying to find a formula to calculate the following sum: $$\sum_{x=0}^n (2ax - {1 \over 2}a^2 - {1 \over 2} a) $$ where $$ a = \left\lfloor {x \over \phi^2} \right\rfloor $$ and $$\phi = {1 + ...
1
vote
1answer
36 views

Finding the integral for function with floor

I am trying to find the integral $$\int_0^n f(x)dx$$ where $$f(x) = 2ax - {1 \over 2}a^2 - {1 \over 2} a $$ and $$ a = \left\lfloor {x \over \phi^2} \right\rfloor $$ and $$\phi = {1 + \sqrt 5 ...
2
votes
2answers
55 views

Problem with Floor Function

I have $f_k(x)=kx-\lfloor kx \rfloor$, where $k\in \mathbb N$ and $x\in\{0,1\}$ and $x\in \mathbb Q$. When I plug in some numbers it seems obvious that $$ f_k(x)+f_k(1-x)=1 $$ for example $k=28, ...
0
votes
1answer
16 views

Remainder operation in terms of the floor function

I came across this identity $$a\bmod{n}= a - \left\lfloor \frac{a}{n} \right\rfloor \times n$$ I see that it works, but I'm struggling to prove it, so I thought I would ask you guys.
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0answers
32 views

summation involving floor function with powers

I want to perform $\sum_{i=1}^k \lfloor{\frac{N}{2^i}}\rfloor7^{i-1} $. Is there any technique which performs it? THanks
3
votes
1answer
88 views

What are these numbers?? (floor(a)=0)

I was just so I decided to go and look up the roots of floor(a) in WolframAlpha where a is any number, real or complex, and of course the interval [0,1) showed up as an answer but I also got these ...
0
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1answer
66 views

How do I solve this summation [duplicate]

$$ \sum_{b=1}^{n} \lfloor\frac{n}{b} \rfloor $$ I can't figure out how to convert this into a closed form. Please help Thanks! Edit: I often come across summations I'm unable to solve. Is there ...
0
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0answers
44 views

Given a non-negative integer $m$ and a positive integer $n$, calculate $\lfloor \frac{m}{n} \rfloor$

Here is the problem: I have a non-negative integer $m$ and a positive integer $n$ I would like to calculate $\lfloor \frac{m}{n} \rfloor$, $\lceil \frac{m}{n} \rceil$ and $m \bmod n$ But I want to ...
-1
votes
2answers
69 views

Given $\lceil x−1 \rceil$, how can I compute $\lfloor x \rfloor$ without using $modulo$?

I have $\lceil x \rceil = -\lfloor -x \rfloor$, but I can't figure out how to rely on this in order to get $\lfloor x \rfloor$ from $\lceil x−1 \rceil$. For the record, I am only interested in ...
0
votes
2answers
83 views

Establish a trigonometry-based $floor$ function

I have established the following function for calculating $floor$: $$f(x)=x-\frac{1}{2}-\frac{\arcsin(\sin(\pi(x-\frac{1}{2})))}{\pi}$$ It works correctly for all real values in the range ...
0
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0answers
27 views

Inequality with nesting floors and ceilings

EDIT: I changed the inequality to a simpler more indicative format. I need to solve the following inequality for $x$ (find the minimum value of $x$) but I have trouble removing the ceiling and floors ...
2
votes
4answers
96 views

Find the number of elements in the range$ f(x) =[x] + [2x] +[2x/3] +[3x] +[4x] +[5x]$ for $0\le x \le3$.

Find the number of elements in the range $f(x) =[x] + [2x] +[2x/3] +[3x] +[4x] +[5x]$ for $0\le x \le3.$ I cant understand...It will go very long if i keep breaking them into small intervals .
6
votes
3answers
248 views

How do I evaluate this sum(involving the floor function)?

$$ \sum_{i=1}^N\left\lfloor\frac{N}{i}\right\rfloor $$ Is there a closed form expression to the above sum? (Mathematica doesn't give me anything)
2
votes
2answers
88 views

A mathematical way for defining the Floor and Ceiling functions

Given: $Floor(x)=\lfloor x \rfloor$ $Ceiling(x)=\lceil x \rceil$ Where $x$ is a real number. Is there any other (mathematical) way for defining $Floor(x)$ and $Ceiling(x)$? Restrictions: Do ...
1
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2answers
52 views

$\lfloor (25x-2)/4 \rfloor =(13x+4)/3$

This problem is designated for those who have the basic knowledge of floor brackets I know the answer will be 6. But tell me how.
2
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0answers
29 views

Solving an inequality involving sum of floors

Suppose I want to find $t_{critical}(u)$, the least $t\in\mathbb{R}^+$ for a given $u\in\left(0\ldots\dfrac{1}{s}\right]$ satisfying $$f(t)=\lfloor rt\rfloor x+\lfloor s (t-u)\rfloor y + y > h$$ ...
0
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2answers
37 views

floor function identity

I would appreciate if somebody could help me with the following problem: Q: How to proof? $(n,k\in\mathbb{N})$ $$1=\left\lfloor\frac{n}{k}\right\rfloor-\left\lfloor\frac{n-1}{k}\right\rfloor$$
2
votes
1answer
69 views

A general method to efficiently calculate the floor of an element of $\mathbb{Q}[\sqrt{2}]$

I had to decide whether to post this question here on the Mathematics Stack Exchange or on Stack Overflow, but I decided that the question was essentially a mathematical one despite being inspired by ...
1
vote
1answer
49 views

Equation with floor function

How would one solve an equation with a floor function in it: $$a - (2x + 1)\left\lfloor{\frac {a - 2x(x + 1)}{2x + 1}}\right\rfloor - 2x(x + 1) = 0$$ $a$ is a given and can be treated as a natural ...
0
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1answer
30 views

Limit of floor function and sine function

for $$\lim\limits_{x\to k}\lfloor x\rfloor sin\frac{\pi x}2$$ find the limit for $k=0,1,2,3$ i started with $$x-1<\lfloor x\rfloor\le x$$ $$\Downarrow$$ $$xsin\frac{\pi x}2-sin\frac{\pi ...
1
vote
1answer
52 views

Solving equation involving the ceiling function

How can I solve the equation $$\lceil \log_{b}{1024} \rceil = n$$ where $n \in \mathbb{N}$ in terms of $b$? I have seen equations of a similar form (Solving an equation with floor function before), ...
4
votes
2answers
66 views

Finding the summation of the floor of the series identity

I would appreciate if somebody could help me with the following problem: Q: How to proof ? The number of positive divisors of $n$ is denoted by $d(n)$ ...
1
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0answers
56 views

Another functional equation

I would like to find all continuous functions $f \, : \, \mathbb{R} \, \longrightarrow \, \mathbb{R}$ such that : $$ \forall x \in \mathbb{R}, \; f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor \tag{1}$$ ...
1
vote
1answer
70 views

Proof that $ [2x] + [2y] \ge [x]+ [x+y] + [y]$?

I have to proof (or disprove) the following $ [2x] + [2y] \ge [x]+ [x+y] + [y]$ for $x,y \in \mathbb{R}$. [x] and [y] means the floor-function. Can I do the following? (1) Assume $x,y \in [0,1]$, ...