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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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Reasoning about inequalities involving floor functions

I am working on the beginning of an inductive argument and I wanted to confirm that my base case is sound. Let $f(x) = \lfloor x\rfloor - \left\lfloor\dfrac{x}{2}\right\rfloor$ where is $x$ is a ...
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2answers
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Showing $\int_{a}^{b} \left\lfloor x \right\rfloor dx + \int_{a}^{b} \left\lfloor -x \right\rfloor dx=a-b$

I want to show $$\int_{a}^{b} \left\lfloor x \right\rfloor dx + \int_{a}^{b} \left\lfloor -x \right\rfloor dx=a-b$$ I know that \begin{equation} \left\lfloor -x \right\rfloor = \begin{cases} -\...
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Doubt on Integration

$$\int_0^4 \lfloor x/2 \rfloor \ d(x-\lfloor x \rfloor)$$ I don't get how we convert the given differential element into normal dx differential element. I plotted the graphs of $$\lfloor x/2 \...
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1answer
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Proving $\int_0^{n^2} \left\lfloor\sqrt t\right\rfloor \,dt=\frac{n(n-1)(4n+1)}{6}$

I want to prove $$\int_{0}^{n^{2}} \left\lfloor \sqrt{t} \right\rfloor \,dt=\frac{n(n-1)(4n+1)}{6}$$ Is it correct to say that $\left\lfloor \sqrt{t} \right\rfloor=\sqrt{(k-1)}$ and $(k-1)^{2} &...
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Values of $x$ satisfying $\lfloor{x^2\rfloor}=\left(\lfloor{x\rfloor}\right)^2$

Prove that values of $x$ satisfying $\lfloor{x^2\rfloor}=\left(\lfloor{x\rfloor}\right)^2$ is $(0,\sqrt{2})\cup \mathbb{Z}$ My try: Its trivial that every integer satisfies the given equation. Now ...
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Algorithm to find floors of multiples of the golden ratio

What is an algorithm to calculate $\lfloor n\phi \rfloor$ given some integer $n$, where $\phi$ is the golden ratio? I am thinking the easiest way will involve calculating multiples of its continued ...
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For $x \in R$ Prove $\lfloor x - 2 \rfloor = \lfloor x \rfloor - 2$

I was hoping to validate my proof based on the answers to this question: For all real numbers $x$, prove $\lfloor x - 2\rfloor = \lfloor x\rfloor - 2$ Let $x = n + \epsilon $, where $n$ is an ...
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1answer
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Graph of $f(x)=x-[x]-\dfrac{1}{2}$

I have a question regarding this graph, for $f(x)=x-[x]-\dfrac{1}{2}$, where $[\cdot ]$ denotes the greatest integer function. My question is about the graph, why does the first slop has an open ...
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1answer
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Formula containing floor functions. [closed]

How can I solve an equation with multiple floor functions added together? $$ 18 + \lfloor 2.6 \rfloor + \lfloor x \rfloor + \left\lfloor \frac x4 \right\rfloor + 5 = 1 $$
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Let $n$ be an integer, prove that $\lfloor n/2 \rfloor \geq (n-1)/2$

So far, I used the definition of floors to provide an interval. Then I did some algebra in order to get $(n-1)/2$. And I am able to deduce that $\lfloor n/2 \rfloor > (n-1)/2$, but now I'm stuck on ...
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Calculate the floor-function of the expression

Let n be a positive integer and x a real number with x $\ge$ $\frac {3n^2+1}{3}$. Calculate $\lfloor \sqrt{x^2-nx}+\sqrt{x^2-n^2}+\sqrt{x^2+n^2}-3x \rfloor$, where $\lfloor t \rfloor$ is the usual ...
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Calculating the Mean of a Range with Floor and Ceiling Functions?

I did a survey a couple months back, and one of the questions required a range of numbers. I may have discovered my own formula for how to calculate the mean of a range of numbers, but I don't know ...
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Floor Summation Closed Form?

Let $ a_1,a_2,a_3,...a_n $ be a set of positive integers. Does there exist any closed form for the approximation of the sum $$ \sum_{i=p}^n\operatorname{floor}\left(\frac{a_p}{i-p+1}\right) ?$$ If ...
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Find sum $ \sum\limits_{k=2}^{2^{2^n}} \frac{1}{2^{\lfloor \log_2k \rfloor} \cdot 4^{\lfloor \log_2(\log_2k )\rfloor}} $

Calculate sum $$ \sum_{k=2}^{2^{2^n}} \frac{1}{2^{\lfloor \log_2k \rfloor} \cdot 4^{\lfloor \log_2(\log_2k )\rfloor}} $$ I hope to solve this in use of Iverson notation: my try $$ \sum_{k=2}^{2^{2^...
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Reasoning with fractional parts and the Möbius function.

Let $S(p_k,x)$ be the set of all elements $s$ where $s \le x$ and gcd$(s,p_k\#)=1$ where $p_k$ is the $k$th prime and $p_k\#$ is the primorial for $p_k$. Let $|S(p_k,x)|$ be the count of elements in $...
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Floor function of sum of square roots

$f(x)=\sqrt{x^{2}-10x+314}+\sqrt{x^{2}+20x+325}$. Find the minimum of $\lfloor{f(x)}\rfloor$. So this becomes $f(x)=\sqrt{(x-5)^{2}+17^{2}}+\sqrt{(x+10)^{2}+15^{2}}$, and simply by putting in $x=0$, ...
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Find range of $x$ satisfying $\left \lfloor \frac{3}{x} \right \rfloor+\left \lfloor \frac{4}{x} \right \rfloor=5$

Find range of $x$ satisfying $$\left \lfloor \frac{3}{x} \right \rfloor +\left \lfloor \frac{4}{x} \right \rfloor=5$$ Where $\lfloor\cdot\rfloor$ is the floor function My try: As far as domain of ...
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Calculate the sum $S_n = \sum\limits_{k=1}^{\infty}\left\lfloor \frac{n}{2^k} + \frac{1}{2}\right\rfloor $

I am doing tasks from Concrete Mathematics by Knuth, Graham, Patashnik for trainning, but there are a lot of really tricky sums like that: Calculate sum $$S_n = \sum_{k=1}^{\infty} \left\lfloor \frac{...
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1answer
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A doubt about greatest integer function

This is a question with reference to a previous post. Inequalities on greatest integer function In the question posted there, I understand why the options A, B, D are true or false. But there is no ...
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2answers
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Finding and proving a closed form formula for a recursive formula with floor and ceiling functions

I have $T:$ $\mathbb{N} \rightarrow \mathbb{N}$ Such that $T(1)=1$, $T(n)=T(\lfloor n/2 \rfloor) + T(\lceil n/2 \rceil)$ for all $n\ge2$. My work: If $n$ is even then $\lceil n/2 \rceil = \lfloor ...
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Prove the following result

Prove that if $p$ is a prime number, then p divides $\binom{n}{p} − \lfloor\frac{n}{p}\rfloor$, for all $n > p$. (where the $\lfloor\frac{n}{p}\rfloor$ denotes the greatest integer less than or ...
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1answer
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How to prove $\lceil \log_2{(n+1)} - 1 \rceil \ge \lfloor \log_2(n) \rfloor$? [closed]

Suppose $n$ is a positive integer. How can one show that $\lceil \log_2{(n+1)} - 1 \rceil \ge \lfloor \log_2(n) \rfloor$ ?
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Find $\lim\limits_{x\to0^+}x(\lfloor \frac{1}{x}\rfloor+\lfloor \frac{2}{x}\rfloor+\cdots+\lfloor \frac{k}{x}\rfloor), \, k \in \mathbb N$.

$$M:=x\left(\left\lfloor \frac{1}{x} \right\rfloor+\left\lfloor \frac{2}{x}\right\rfloor+\cdots+\left\lfloor \frac{k}{x}\right\rfloor\right),\, k \in \mathbb N.$$ Using $\lfloor y \rfloor=y-\{y\}$, ...
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2answers
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Calculate the derivative $f(x)=\lfloor x\rfloor(\sin(\pi x))^{2}$

I have a problem with this task because answer which I have does not match the right answer and I don't know where is a mistake.My try:For $x\in \mathbb Z$ $f'_{+}(x)=f'_{-}(x)=0$ so $f'(x)$ exist For ...
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2answers
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Solving a tough equation involving integer functions

I am stuck on solving the equation, given $k\lt\frac{n}{2},\ n,k\ge3$: $$ m=\lceil 2k-\frac{2}{n}\displaystyle\left(\lfloor\frac{n-\lfloor\frac{n}{k+1}\rfloor}{2}\rfloor\right)(k+1)\rceil$$. I think ...
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2answers
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Prove $\lfloor x + n \rfloor= \lfloor x\rfloor + n : n \in \mathbb{Z}$

Prove $\lfloor x + n \rfloor= \lfloor x\rfloor + n : n \in \mathbb{Z}$ So far I have used that $\lfloor x + n \rfloor - n \leq x < \lfloor x + n \rfloor - n + 1$, but I don't know how to continue.
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Recurrence from closed form for $s_{a,b}(m)=(m-1)s_{a,b}(m-1)+s_{a,b}(m-2), s_{a,b}(0)=a, s_{a,b}(1)=b$

We have for $m>1$ $$s_{0,1}(m)=\sum\limits_{k=0}^{\left\lfloor{\frac{m-1}{2}}\right\rfloor}\binom{m-k-1}{k}\frac{(m-k-1)!}{k!}$$ $$s_{1,0}(m)=\sum\limits_{k=0}^{\left\lfloor{\frac{m}{2}}\right\...
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Double summation with floor bounds

I need to express following double sum in terms of $p$ for $p\equiv 0 \bmod 12$: $\sum_{k=0}^{\binom{p}{2}} \sum_{l=0}^{\lfloor{\frac{(p-2)k}{3}}\rfloor} \lfloor{\frac{(p-3)l}{4}}\rfloor$ I tried ...
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3answers
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Find all $n$ $\in$ $\Bbb Z^+$ such that: $\lfloor\frac{n}{2}\rfloor \cdot \lfloor \frac{n}{3} \rfloor \cdot \lfloor \frac{n}{4} \rfloor = n^2$

Find all the numbers $n$ $\in$ $\Bbb Z^+$ such that: $$\left\lfloor\frac{n}{2}\right\rfloor \cdot \left\lfloor \frac{n}{3} \right\rfloor \cdot \left\lfloor \frac{n}{4} \right\rfloor = n^2$$ I never ...
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Finite double sum $\sum_{k=0}^N\sum_{l=0}^M\left\lfloor\frac{k+l}{c}\right\rfloor$; any advanced summation technique?

Let $M,N,c$ be positive integer. It was astonishing when trying to solve $\sum_{k=0}^N\sum_{l=0}^M\left\lfloor\frac{k+l}{c}\right\rfloor$ to obtain this rather complex looking result \begin{align*} ...
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1answer
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Integral $\int_{-3}^2 \frac{[x]}{[x]^2+2[x]+x^3}dx$

I came across this question which was supposed to be solved in about $1$ or $2$ minutes, but I came across a severe roadblock. The question was: Integrate $$\int_{-3}^2 \frac{\lfloor x\rfloor}{\...
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1answer
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Find out the value of the integral $\int_{-2}^{2} \lfloor x^2-1\rfloor dx$

Find out the value of the integral $$\int_{-2}^{2} \lfloor x^2-1\rfloor dx$$ where $[x]$ denotes the floor function (i.e., $[x]$ is the greatest integer $\le x$.) My attempt ..... $$\int_{-2}^2 \...
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2answers
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Finding $\lim\limits_{x\to 0} \frac{\lfloor x \rfloor}{x}$ and the different definitions of fractional part function.

I understand that $\lim\limits_{x\to 0} \frac{\lfloor x \rfloor}{x}$ does not exist because RHL is $0$ and LHL is $\infty$. However, when I tried to calculate the limit of the equivalent expression $1-...
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Why $ \int_{0}^{\infty} \lfloor x \rfloor x dx= \sum_{0}^{\infty}n\int_{n}^{n+1}xdx $

I am trying to learn how to work with the floor function, and I would really appreciate if someone can explain me the reason behind: $ \int_{0}^{\infty} \lfloor x \rfloor x dx= \sum_{0}^{\infty}n\...
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How do I find $E\lfloor X \rfloor$ when $X \sim \text{exp}(2)$?

I am trying to learn from my mistakes, and faced the following problem: Let $X \sim \exp(2)$ and $Y=\lfloor X \rfloor$, compute $E[Y]$. Well my false attempt was: First compute the PDF of Y: $ ...
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1answer
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Prove/disprove that $\lfloor x\rfloor \leq t \iff x\leq\lfloor t\rfloor +1$

Prove/disprove that $\lfloor x\rfloor \leq t \iff x\leq\lfloor t\rfloor +1$ Playing around I can see why this is true, but I have no idea how to prove that, any ideas?
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2answers
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Rigorous or not?

I want to prove $$T(n,k)=\frac{n}{\left\lfloor\frac{n+k+1}{2}\right\rfloor}(T(n-1,k)+T(n-1,k-1))=\binom{n}{k}\binom{n-k}{\left\lfloor\frac{n-k}{2}\right\rfloor}$$ First we know only that $$T(n,k)=0, n&...
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1answer
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Express $\sum_{a=1}^{p-1} \lfloor{(v+qa)/p}\rfloor$ in closed form.

Here $p$ and $q$ are primes but likely not necessary for the answer. Also $p < q$ and $v\in\left\{{0,1,\dots,p*q-1}\right\}$. This problem arises from counting the number of reducible quadratics ...
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2answers
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Inverse function of $x-\lfloor x \rfloor $ and $(x-\lfloor x \rfloor)^2$

I need to find the inverse of these two functions if they exist: $$f_1 = x-⌊x⌋, 1\leq x<2, 0\leq f_1<1$$ and $$f_2 = (x-⌊x⌋)^2, 1\leq x<2, 0\leq f_2<1$$ I worked through it and I think ...
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4answers
86 views

On a multisection of a series

Let $x$ be a real number and $\lfloor{x}\rfloor$ denote the greatest integer less than or equal to $x$. Let $r$ be a positive integer. Let $a$ and $n$ be nonnegative integers such that $n-a+1\geq r$. ...
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1answer
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A series question whose indices involves greatest integer function

Let $x$ be a real number and $\lfloor{x}\rfloor$ denote the greatest integer less than or equal to $x$. Let $a$ and $n$ be nonnegative integers such that $n\geq a+1$. Prove that $$ \sum_{k=a}^{n} f(k)...
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5answers
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$\left\lfloor \frac{a-b}{2} \right\rfloor + \left\lceil \frac{a+b}{2} \right\rceil = a$ when $a,b$ are integers? [closed]

Let $a$ and $b$ be positive integers. If $b$ is even, then we have $$\left\lfloor \frac{a-b}{2} \right\rfloor + \left\lceil \frac{a+b}{2} \right\rceil = a$$ I think the equality also hold when $b$ ...
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1answer
78 views

Maximum value of sum with integer parts

How can I find the max value of $\sum_{i}^{n} \lfloor\frac{i}{k}\rfloor (k-1)$ with $k$ integer $\in [1,100]$? I can express this sum in terms of n and k (it's quite easy) but I'm not able to find its ...
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2answers
71 views

Prove that $\sum\limits_{k=1}^n\lfloor n/k\rfloor+\lfloor \sqrt{n} \rfloor$ is even. [duplicate]

Let $n$ be a natural number. Prove that $\displaystyle \sum_{k=1}^n\lfloor n/k\rfloor+\lfloor \sqrt{n} \rfloor$ is even. I tried to introduce the fractional part, but it didnt help me. Next, I ...
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0answers
50 views

Prove the equality $[\sqrt n+\sqrt{n+1}]=[\sqrt{4n+2}]$. [duplicate]

I'm reading Apostol Analytic Number Theory, the question ask me to prove for positive integer $n$, $$[\sqrt n+\sqrt{n+1}]=[\sqrt{4n+2}].$$ My approach is, define $f(n)=[\sqrt n+\sqrt{n+1}]$, after ...
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0answers
35 views

Lebesgue integral of integer part function

let $\lambda $ be a Lebesgue measure and $[x]$ is the integer part of $x$ \begin{cases} f(x)= \frac{1}{[x]}, & x \geq0 \\ f(x)=0, & \text{otherwise} \end{cases}$$ My question is how I can ...
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1answer
70 views

How do we prove that $[n+1/2]+[n+2/4]+[n+4/8]+[n+8/16] …=n$? [closed]

Would someone please help me to prove the above relation? $[.]$ denotes the floor function $n$ is a positive integer.
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1answer
31 views

Equivalence between ceil and floor functions

I was reading heap data structures from various sources. They used to explain heap as stored in array. One source has array starting at index 0. Other has it starting at 1. They specify different ...
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0answers
34 views

Proving multiplication floor functions

If x, y is bigger than zero (I mean real numbers bigger than 0.) Then why does this always works? [x][y]<=[xy]<([x]+1)([y]+1) <= means less or equal to [] brackets are floor function
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3answers
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Why is $\left\lfloor\frac{\lfloor a\pi\rfloor}{a}\right\rfloor=3$ for $a>0$?

Why is this true? $$\left\lfloor\frac{\lfloor a\pi\rfloor}{a}\right\rfloor=3 \text{, for } a>0$$ I need this to solve the Ukraine Math Olymipiad 1999. "$\lfloor\cdot\rfloor$" indicates the floor ...