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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Greatest Integer Function linear equation

Given that $2[x]=x+2(x)$, $[x]$ if the Greatest Integer Function and $(x)$ is the fractional part of $x$, find the value (s) of $x$. I tried replacing $(x)=x–[x]$ but for an equation in $x$ and $[x]$....
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Evaluate $\int_{1}^{n} \lfloor x \rfloor^{x- \lfloor x \rfloor} dx$.

$\int_{1}^{n} [x]^{x-[x]} dx$ I tried to approach this with riemann sum method but it seems impossible by that way. Even using other general integration techniques it seems quite complicated .I have ...
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1answer
73 views

How do limits work with floor/ceiling?

I'm interested in the below equation: $$\frac{n}{\operatorname{floor}(\frac{x}{n})}$$ Plotting with $n = 1..100$ shows the graph being slightly more aliased as $n$ increases and a discontinuity ...
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1answer
22 views

Find the solution of $1 \le \left\lfloor \left\lvert \frac{x+1}{x-3}\right\rvert +x\right\rfloor \lt 2$

Find the solution of $$1 \le \left\lfloor \left\lvert \frac{x+1}{x-3}\right\rvert +x\right\rfloor \lt 2$$ My try: The only thing i know is that $$\left\lfloor \left\lvert \frac{x+1}{x-3}\right\rvert +...
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0answers
68 views

Proving $\lfloor e^x \rfloor =\lfloor e^{\lfloor x \rfloor} \rfloor$

Prove that $$\lfloor e^x \rfloor =\lfloor e^{\lfloor x \rfloor} \rfloor \tag{1}$$ I was actually trying to prove $$\lfloor \sqrt{x} \rfloor=\lfloor \sqrt{\lfloor x \rfloor} \rfloor$$ and i ...
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3answers
62 views

Is $\lfloor2\sqrt{n-\lfloor \sqrt{n} \rfloor}\rfloor <\lfloor2\sqrt{n} \rfloor$ true for $n\geq 0$?

I have reasons to believe that $\lfloor2\sqrt{n-\lfloor \sqrt{n} \rfloor}\rfloor <\lfloor2\sqrt{n} \rfloor$ for $n\geq 0$. How could I go about proving (or disproving) this?
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1answer
44 views

I need to solve one equation, but I dont know how to solve equation with floor functions

Im a student with not such a knowledge to solve equations with floor functions. I want to ask, if it is even possible and if it so, how is possible to prove this equation to be true. ...
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24 views

Differentiation wrt a Floor Division

I have been racking my brain on how i should go about resolving: $\frac{d( n (\frac{g}{n} - \lfloor\frac{g}{n}\rfloor))}{d \lfloor\frac{g}{n}\rfloor}$ For this case replace ${\lfloor\frac{g}{n}\...
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8answers
89 views

Proving that the $\lfloor-x\rfloor= -\lceil x\rceil$

My homework assignment has asked me to prove that $\lfloor-x\rfloor = -\lceil x\rceil$. Conceptually this makes perfect sense to me, I just am at a loss for how to start actually proving it. I figure ...
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2answers
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Chebyshev function - show that $\psi(x)>(x-2)\log2-\log(x+1)$

The question I'm trying to do is this: Assume $x>2$ and $n=\lfloor x/2\rfloor$. Show that $\psi(x)>(x-2)\log2-\log(x+1)$, given the inequality $2n\log2-\log(2n+1)<\psi(2n)$. All I've ...
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3answers
62 views

Approximating series of fractions [duplicate]

Let $$ P = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}}+ \frac{1}{\sqrt{4}} ... +\frac{1}{\sqrt{10000}}$$ what is the value of the floor function of P? My try: I tried assuming these 2 bounds $$ P_x =...
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1answer
71 views

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
56 views

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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1answer
67 views

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
57 views

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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2answers
52 views

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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2answers
56 views

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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1answer
49 views

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
39 views

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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1answer
26 views

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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0answers
32 views

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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0answers
28 views

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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0answers
34 views

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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2answers
117 views

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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1answer
57 views

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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2answers
42 views

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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4answers
85 views

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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4answers
159 views

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
42 views

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
56 views

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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2answers
51 views

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
39 views

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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135 views

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
66 views

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
70 views

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
30 views

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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3answers
111 views

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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0answers
73 views

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
71 views

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
45 views

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
82 views

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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2answers
44 views

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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0answers
36 views

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
35 views

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
82 views

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
54 views

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 ...
2
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2answers
54 views

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
88 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
62 views

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)...