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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Can this expression for a smooth triangle wave function be “rotated” about the origin to produce a smooth floor function?

I had come up with an expression for a smooth triangle wave function, below, and wondered if it can somehow be rotated 45 degrees about the origin in order to yield a smooth floor function. $$ TW(x)=...
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How to prove the following floor function inequality?

For any $M\in\mathbb{Z}^+$ and $n\ge\lceil\log_2(M) \rceil, n\in\mathbb{Z}$, we have $$ M^2\log_2(M)\le 4\sum_{i=1}^n\left(M-\sum_{k=0}^{2^{i-1}-1}\left\lfloor\frac{M+k}{2^i} \right\rfloor\right)\left(...
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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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Integral of a Floor Function (and Graph)

Draw the graph of $f(x)$ over the interval $[0,3]$, where $$f(x)=\int_0^x\lfloor t\rfloor^2\mathrm dt.$$ I don't quite know if I need to use this formula $$\int_0^n\lfloor t\rfloor^2\mathrm dt=\frac{...
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Prove that $f(x) = x - {\lfloor}x{\rfloor}$ is periodic.

How do I prove that $f(x) = x - {\lfloor}x{\rfloor}$ is periodic and find its minimal period? I've taken the following steps: Let $x = x_0 + \Delta{x}$ where $x_0 \in \mathbb Z$ and $\Delta{x} \in [...
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Explain the graph of $\lfloor \arctan x\rfloor$

I recently started studying continuity and I tried to figure out continuity of $\lfloor \arctan x\rfloor$ using graphs. I got a similar graph to the one which is in the Desmos graph plotter. To check ...
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Prove property of floor function [duplicate]

Prove that, for any $a, b, c \in \mathbb{N}$, $\bigl\lfloor \frac{a}{bc} \bigr\rfloor = \Bigl\lfloor \frac{\bigl\lfloor \frac{a}{b} \bigr\rfloor}{c} \Bigr\rfloor$. I have no idea how to prove this. ...
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Floor value of sum whose general term satisfy recursive relation.

Consider the sequence $x_{n}$ given by $\displaystyle x_{1} = \frac{1}{3}$ and $x_{k+1}=x^2_{k}+x_{k}$ and Let $\displaystyle S = \frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots \cdots +\frac{1}{x_{2008}}$...
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How to prove that $\sum_{k=1}^{am-1}\left\lfloor\frac ka\right\rfloor=a\sum_{k=1}^{m-1}k$?

Let $n=am+r$ where $0\le r\le a-1$ and $m=\left\lfloor\frac na\right\rfloor$. Of course, by some examples we can see that such a thing is true, but I'm trying to prove that mathematically, simply by ...
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How can $\int_0^x\lfloor t \rfloor^2dt$ be written as $\sum_{j=1}^{\lfloor x - 1 \rfloor} j^2 + q^2r$

Question 6(c) from Section 1.15 Exercises of Apostol's Calculus is the following: Find all $x > 0$ for which $\int_0^x\lfloor t \rfloor^2dt = 2(x-1).$ A particular piece of reference material ...
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How to compute density function of $Y=X-\lfloor X \rfloor$?

We're given that X is a continous r.v. and has a density function of $f_X(x)$ and we're asked to find the density function of $Y=X-\lfloor X \rfloor$ in terms of $f_X(x)$. We're also given the hint : ...
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Question involving greatest integer function

The question $f(x)=[2x+\sqrt n]$, where $[x]$ is the greatest integer less than or equal to $x$ and $n<100$. If $f(x)$ is discontinuous in the interval $[1,1.5)$, then find the total number ...
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How can $\int_{-3}^3\lfloor{x^2}\rfloor dx = 2 \cdot \int_0^3\lfloor{x^2}\rfloor dx$?

This website simplifies the integral $$\int_{-3}^3\lfloor{x^2}\rfloor dx$$ into $$2 \cdot \int_0^3\lfloor x^2 \rfloor dx.$$ How is that done? I'm unsure which property of the integral it is. The ...
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Evaluate $ \lim_{x \to -0.5^{-}} \left\lfloor\frac{1}{x} \left\lfloor \frac{-1}{x} \right\rfloor\right\rfloor$

Evaluate $$L=\lim_{x \to -0.5^{-}} \left\lfloor\frac{1}{x} \left\lfloor \frac{-1}{x} \right\rfloor\right\rfloor $$ My try: Let $t=\frac{1}{x}$ Now when $ t \to -0.5^{-}$ we have $t \to -2^{+}$ we ...
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Finding an inverse for a function with floor

I have the function $f:\mathbb{N}\to \mathbb{N}$ defined by $$f(i) = i\cdot a + \left\lfloor\frac{i\cdot a}{b-1}\right\rfloor$$ for integers $a,b\geq2$, and I'm trying to find an inverse, so that ...
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Can Greatest integer function and limit be Interchanged

Consider the Limit $$ L_1 = \lim_{x \to 0}\left\lfloor\frac{\sin x}{x}\right\rfloor . $$ We have $$ L_1 = \lim_{x \to 0} \left\lfloor \frac{x-\frac{x^3}{6} + \dots}{x} \right\rfloor = \lim_{x \to 0}\...
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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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Continuous antiderivative of $\frac{1}{1+\cos^2 x}$ without the floor function.

By letting $u = 2x$ and $t = \tan \frac{u}{2}$, I found the continuous antiderivative of the function to be: $$\int \frac{1}{1+\cos^2 x}dx\\= \int \frac{2}{3+\cos2x} dx\\ = \int \frac{1}{3+\cos u}du \...
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Evaluate $\lim_{n \to \infty} \sqrt{n^2+2n}-\left[\sqrt{n^2+2n}\right]$

Evaluate $$\lim_{n \to \infty} \sqrt{n^2+2n}-\left[\sqrt{n^2+2n}\right]$$ where $[.]$ is Greatest integer function My try: We have $$L=\lim_{n \to \infty} \left\{\sqrt{n^2+2n} \right\}$$ where $\...
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Infinitely many $n$ such that $2^n$ in base $10$ starts with $7777777$

I'm attempting to prove that there are infinitely many $n$ such that the first $7$ digits in the base $10$ expressions of $2^{n}$ are $7777777$. However, I don't even know where to start. Apparently I'...
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Evaluate $\int_{-n}^{n} (-1)^{\lfloor x \rfloor}dx$

I am not sure how to solve this. I tried checking it for odd/even function but we don't have $\lfloor -x \rfloor=-\lfloor x \rfloor$
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Every function can be represented by a rising sequence of simple functions

I have the following problem: Let $(\Omega,F)$ be a measure space, $f:\Omega\rightarrow\mathbb{R}$, $f_n:\Omega\rightarrow\mathbb{R}$ with $x\mapsto min\{\frac{1}{2^n}\lfloor 2^n f(x)\rfloor,n\}$. ...
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Recurrence problem: find $a_{1000}$ from $a_{0}$

Hi I'm stuck at yet another question. $a_{0}=5$. Given $a_{n+1}a_{n} = a_{n}^{2} + 1$ for all $n \ge 0$, determine $\left \lfloor{a_{1000}}\right \rfloor$. So I got $a_{n+1}=a_{n} + \frac{1}{a_{n}}...
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Simplifying floor > $ \left \lfloor{\frac{\left \lfloor{\frac{n}{2}}\right \rfloor}{2}}\right \rfloor= \left \lfloor{\frac{n}{2^2}}\right \rfloor$

Is the following true $ \left \lfloor{\frac{\left \lfloor{\frac{n}{2}}\right \rfloor}{2}}\right \rfloor= \left \lfloor{\frac{n}{2^2}}\right \rfloor$ such that, $n \in \mathbb{I}$
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A strange mathematical result from greatest integer function.

I have made a function to plot a graph of stairs with heigth '$h$' & length '$l$' ( Desmos link) where I assume that $l$ & $h$ and $x$ are positive, real and not equal to zero(i.e the stairs ...
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How to solve equation with the floor function? 100 sided die Problem

The 100 sided die problem has been asked before: 100-sided die probability. You are given a 100-sided die. After you roll once, you can choose to either get paid the dollar amount of that roll OR ...
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Solving equation with floor

Is it possible to solve for $i$ in the following equation? EDIT- WolframAplha says it is possible but how do I do it? $$\left \lfloor{\displaystyle \frac n{2^i}}\right \rfloor =1 $$ I am not ...
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sums of floor and sqrt functions

Let $n\in\mathbb{N}$ fixed. How do you compute $$\sum_{n^2<m<(n+1)^2}\left\lfloor\frac{1}{\sqrt{m}-\lfloor\sqrt{m}\rfloor}\right\rfloor?$$ I did some examples for values of $n$, but didn't find ...
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Find maximum of a composition of floor functions

Is there a way to find the maximum of a composition of floor functions? $f(x) = \lfloor \lfloor x \times a \rfloor \times b \rfloor \times c \rfloor \times d - x$ $x \in [n, m], n \geq 0, n < m$ ...
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On the validity of a manual-calculation-friendly variation of the Legendre's formula: $\nu_p(n!)=\sum_{i=1}^\infty\lfloor\frac{n}{p^i}\rfloor$.

The Legendre's formula gives $\alpha$ in $$p^\alpha || n!$$ where $p$ is a prime number. To calculate $\nu_p(n!)$ on paper, one should normally find the quotients $q_i$ in these equations by long ...
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Is there any relation between $\lfloor n/k\rfloor$ and $\lfloor n/(k+1)\rfloor$?

Given two integers $n$ and $k$ such that $n\geq k+1$. Can we find any relation between $\left\lfloor\dfrac{n}{k}\right\rfloor$ and $\left\lfloor \dfrac{n}{k+1}\right\rfloor$? At first, I thought ...
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Definite integral concerning the greatest integer function

Evaluate the integral $$I=\int_{0}^{x} \lfloor t+1 \rfloor^3 dt$$where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. The answer is given as $$\Bigg[\frac{\lfloor x \...
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On continuity of floor, modulus and fractional part function.

If $$f(x)=\begin{cases}\dfrac{e^{\lfloor x\rfloor}+|x|-1}{\lfloor x\rfloor+\{2x\}}&,\ x\neq0 \\\ \frac{1}{2}&,\ x=0\end{cases}$$comment on continuity of $f(x)$ at $x=0$. Where $\lfloor .\...
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Function $f:\mathbb{R}\mapsto \mathbb{R}$ satisfying $f(f(x))=x+\lfloor x\rfloor$

I'm looking for a function $f$ from reals to reals such that $$f(f(x))=x+\lfloor x\rfloor$$ ...or, in other words, a functional square root of the function $x+\lfloor x\rfloor$. I've been able to ...
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Concrete Mathematics(3.2) - Doubt in a Summation involving Floor Function

Finding number of winners W This is from Concrete Mathematics Chapter 3 Page 73. I don't understand how they got from step 3 to step 4. I proceeded from step 3 as follows: $$ W = \sum_{k,m} [ k^3 ...
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Summation of the minimum of two options

I am trying to figure out how to sum a minimum functions, i.e. $$\sum_{i=0}^{\lfloor\frac{m}{2}\rfloor} \sum_{j=0}^{\lfloor\frac{n}{2}\rfloor} \min \Bigl( m - 2i, \lfloor\frac{n-2j}{2}\rfloor\Bigr)$$ ...
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Show that $f(x) = \lfloor \sqrt{2} \cdot x\rfloor$ is primitive recursive function.

Trying to wrap my head around primitive recursive functions. Especially bounded minimalisation and how to prove something is indeed primitive recursive. Found the following problem for the subject ...
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Taking the integral with absolute value and the floor function

I have never integrated using the floor function before, so I just need help starting the following problem. $$\int_{3}^{4} \frac{|x-1|}{\lfloor 2x-5 \rfloor} dx =?$$
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Find the range of $a$

If the function $$f(x)=[4.8+a\sin x]$$ where $[.]$ denotes the greatest integer function, is an even function then find the range of $a$. So, if the function of even then $f(-x)= f(x)$. Therefore,$$[...
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Why isn't $\lim_{x \to 0^+} x \left[ \dfrac px \right] = 0$?

Find $L$ $$\lim_{x\to{0^+}} x\left(\left[\frac 1x\right] + \left[\frac 2x\right] + \left[\frac 3x\right] + \cdots + \left[\frac {12}x\right]\right) = L$$ Here $[t]$ represents the greatest ...
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Is it true that $[N*a]=N*[a]$?

Let $[\cdot]$ denote closest integer of a real number. Is it true that $$ [N*a]=N*[a] $$ where $N\in \mathbb{N}$ and $a\in \mathbb{R}$?
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Sum of $34$ Terms Involving Floor Function

Find the value of $$\left\lfloor \frac{18}{35}\right\rfloor +\left\lfloor \frac{18\cdot 2}{35}\right\rfloor +\left\lfloor \frac{18\cdot 3}{35}\right\rfloor +\cdots \cdots +\left\lfloor \frac{18\cdot ...
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Find the real values of $x$ that satisfy the equation $7[x]+23\{x\}=191$

For any real number $x$, $[x]$ denotes the largest integer less than or equal to $x$ (i.e. floor function) and $ \{x\}=x-[x]$ .Then, the number of real solutions of the equation $$7[x]+23\{x\}=191$$ ...
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How to find continuity and differentiability of a function which includes Greatest Integer Function?

Lets consider function $ f(x) = [x] $ where $[\space]$ denotes a greatest Integer function. let $ g(x)$ be an arbitrary function. How do we determine whether the function $f(g(x))$ and $g(f(x))$ is ...
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Find all integral ordered pairs $(n,k)$ such that $\left\lfloor\frac{n^2+18n+10}{2}\right\rfloor = k^2$.

I had two problems that I want to solve. The first one was easy, but the second one... not so much: First Problem: Find all values of $n$ such that$$\frac{(n+1)(n+9)+8n+1}{2} = n^2.$$ ...
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What is the derivative of this function: $\frac {d}{dx}x^{\lfloor{x}\rfloor}?$

What is the derivative of the following function? $$\frac {d}{dx}x^{\lfloor{x}\rfloor}$$ Here, $\lfloor x \rfloor$ is the floor function. I tried: $$\frac {d}{dx} x^x=\frac {d}{dx} e^{x \ln x}=x^...
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If $a_1\le a_2 \le \dots \le a_n$ and $a_1 + 2a_2 + \cdots + na_n = 0$, then $a_1[x] + a_2[2x] + \cdots + a_n[nx] \geq 0$

Consider a positive integer $n$ and the real numbers $a_1 \leqslant a_2 \leqslant \cdots \leqslant a_n$ such that $\displaystyle a_1 + 2a_2 + \cdots + na_n = 0$ Prove that $\displaystyle a_1[x] +...
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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 ...
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$\lim_{n\to\infty}\frac{n -\big\lfloor\frac{n}{2}\big\rfloor+\big\lfloor\frac{n}{3}\big\rfloor-\dots}{n}$, 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 ...