Questions tagged [ceiling-and-floor-functions]
This tag is for questions involving the greatest integer function (or the floor function) and the least integer function (or the ceiling function).
2,411 questions
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Prove that $\sum_{k=1}^{n} \left\lfloor \log_{2}\!\left(\frac{2n}{2k-1}\right) \right\rfloor = n$ where $n$ is a natural number.
How can I prove that $\sum_{k=1}^{n} \left\lfloor \log_{2}\!\left(\frac{2n}{2k-1}\right) \right\rfloor = n$, where $n$ is a natural number?
I discovered this identity while trying to prove Prove using ...
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Determine $x,y \in \Bbb{N}$ with the property that $[\sqrt[n]{x^n+y}]=[\sqrt[n]{y^n+x}]$
the problem
Let $n \in \Bbb{N}, n\geq 2$. Determine $x,y \in \Bbb{N}$ with the property that $[\sqrt[n]{x^n+y}]=[\sqrt[n]{y^n+x}]$, where $[a]$ represents the integer part of the real number $a$.
my ...
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If $2^k < n^m-1$ it means that expression is $2^{k-1}$
If $n,m,k \in \mathbb{N}$ must be such that $2^k < n^m-1$ it follows that $$\left\lfloor \dfrac{(n^m+1)^{k+1}}{n^{2m}-1}\right\rfloor-(n^m+1)\left\lfloor \dfrac{(n^m+1)^k}{n^{2m}-1}\right\rfloor = ...
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Find the whole part of $E(n)=(\sqrt[3]{n+1}-\sqrt[3]{n}+\sqrt[3]{n-1})^3, n \in \mathbb{Z}$
The Problem
Find the whole part of $E(n)=(\sqrt[3]{n+1}-\sqrt[3]{n}+\sqrt[3]{n-1})^3, n\in \mathbb{Z}$ Do the same thing for $\sqrt{}$ instead of $\sqrt[3]{}$
My Idea
Use the identity
$$(x+y+z)^3=x^3+...
- 413
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Intervals of continuity of $ f(x)=\frac{\Gamma(x+1)}{\lfloor\Gamma(x+1)\rfloor+1}$
Let
$$\Gamma(x)=\int_0^\infty t^{x-1}e^{-t}\,dt.$$
Define
$$f(x)=\frac{\Gamma(x+1)}{\lfloor\Gamma(x+1)\rfloor+1},\qquad x\in[0,\infty).$$
Find the maximal open intervals $I_n=(a_n,b_n)\subset[0,\infty)...
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(INMO 2009) Find all real numbers $x$ such that $\lfloor x^2 + 2x\rfloor = \lfloor x\rfloor^2 +2\lfloor x\rfloor$.
First of all, the notation $\lfloor x\rfloor$ is the greatest integer not exceeding $x$. That is $\lfloor x\rfloor = \max\{m \in \mathbb{Z}: m\leq x\}$.
Question: Find all real solutions of $\lfloor x^...
user1598427
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Does every $r \geq 1$ give a different power sequence? [closed]
Let $r$ be a real number greater than or equal to $1$. I define the positive integer power sequence associated with $r$ to be the sequence of floors of $n^r$, where $n$ is a positive integer. For ...
- 24.8k
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solution-verification | determine the number of solutions of $[ \sqrt{k} ] + [ \sqrt{k+1} ] =n $
the problem
Let $n\in \Bbb{N}$ Determine how many numbers $k\in \Bbb{N}$ verify $[ \sqrt{k} ] + [ \sqrt{k+1} ] =n $
my idea
So, for a start, I use the fact that every natural number can be bonded ...
- 413
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2
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418
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the infinitude of perfect squares in a sequence
Let $\alpha > 1$ be an irrational number such that $\alpha^k \notin \mathbb{Q}$ for every $k \geq 1$.
Prove or disprove: there exists $n \in \mathbb{Z} \setminus {0}$ for which
$$
a_k(n) = \lfloor \...
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Solving the recurrence $ a(n)=a(n-4)+\left \lfloor{\frac{(n+3)^2+8}{24}}\right \rfloor $, with $a(1)=a(2)=a(3)=1$ and $a(4)=2$
I need help solving the recurrence relation
$$
a(n)=a(n-4)+\left \lfloor{\frac{(n+3)^2+8}{24}}\right \rfloor
$$
with starting values $a(1)=a(2)=a(3)=1$ and $a(4)=2$.
I am looking for closed form ...
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Show that $|S_1|+|S_2|+\dots+|S_m|=[\frac{m+1}{2}][\frac{m+2}{2}]$ for any $m\in\mathbb N^*$
The problem
For each $n \in \mathbb N^*$ we denote $S_n=\{ x\in[0,1)| x\{nx\} = \frac{1}{2} \}$. Let $|S_n |$ be the number of elements of the set $S_n$. Show that $|S_1|+|S_2|+\dots+|S_m|=[\frac{m+1}{...
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Find the value of the floor function of $1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{28}}$ [duplicate]
The problem
Find the value of the floor function of $1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{28}}$.
My idea
I tried the idea in this post, but got that the number is between $8....
- 1,199
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Evaluate $\int_{0}^{1}\Big\lfloor{\sqrt{1+\frac{1}{x}}\Big\rfloor}dx$
I need help in evaluating the above integral.
I know how to evaluate the integral $$\int_{0}^{1}\left(\sqrt{1+\frac{1}{x}}\right)dx$$
Let
$$I=\int_{0}^{1}\left(\sqrt{1+\frac{1}{x}}\right)dx$$
$$\...
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Confusion about whether $\lfloor 4.\overline{9}\rfloor =5$ or $\lfloor 4.\overline{9}\rfloor =4$ [closed]
I mean, I get it but this seems like such a trick question. Will it be $\lfloor 4.\overline{9}\rfloor=5$? Or just like how it seems intuitively $\lfloor 4.\overline{9}\rfloor=4$? This is my first post....
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Really simple ceiling inequality.
For rational numbers $x$ and $y$, I'm pretty sure that $\lceil x - y \rceil \geq \lceil x \rceil - \lceil y \rceil$, but I'm not quite confident in my proof. Also, it feels like there might be a ...
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Does there exist a floor function for $\Bbb{C} = \{a + bi :a,b \in \Bbb{R}\}$ since there exists one for $\Bbb{R}$? Namely it round to $\Bbb{Z}[i]$
I did some preliminary research on this, and google links say it can be done with the "nearest Gaussian integer". But what about something more like $\left[z\right] = [x] + [y]i$? Where $[\...
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Prove that $\left\lfloor \frac{n}{k+1+M+1} \right\rfloor\geq \frac{n}{2k+2M+2} $
Assuming that $n\geq 2k+2M+4$ and $M,k \geq 1$ and $n,M,k\in\mathbb{N}$, I want to show that
$$\left\lfloor\frac n{k+1+M+1} \right\rfloor \ge\frac n{2k+2M+2}.$$
In my idea I started with the ...
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Graph of [y] = [x] i.e. floor(y) = floor(x)
Actually I wanted to solve this equation where x and y are independent variables. Like kind of obtain the solution as we obtain in case of absolute value function. |x| = |y| which is y = x or -x.
My ...
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Evaluating $\lim_{x \to \infty}{\frac{\lfloor ax \rfloor}{x}}$
I'm reading A course in Real analysis by Junghenn, and the book asks the question
Evaluate limit without using l'Hospital's rule $\forall a > 0$,
$\lim_{x \to \infty}{\frac{\lfloor ax \rfloor}{x}}$...
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Is $f(n) = \lfloor n \sin n \rfloor$ an $\mathbb{N} \to \mathbb{Z}$ surjection?
Is the function $f \colon \mathbb{N} \to \mathbb{Z}$ defined by $f(n) = \lfloor n \sin n \rfloor$ surjective?
In other words, given $m \in \mathbb{Z}$, we seek $n \in \mathbb{N}$ such that
$$
\frac{m}{...
- 3,145
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Let $b(n) = \min_{k \in \mathbb{Z}^+} \left( k + \frac{n}{k} \right)$. Prove that $\lfloor b(n) \rfloor = \lfloor \sqrt{4n + 1} \rfloor$.
PUTNAM 1973 A3
$n$ be a fixed positive integer and let $b(n)$ be the minimum value of $k + \frac{n}{k}$, where $k$ is allowed to range through all positive integers. Prove that $\lfloor b(n) \rfloor =...
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How to evaluate $\sum_{k=1}^{\infty} \left( \left\lfloor \sqrt[n]{k} \space \right\rceil \right)^{-a} $?
Motivation:
I recently encountered the problem
$
\sum_{k=1}^{\infty} \frac{1}{\left\lfloor \sqrt[4]{k} \right\rceil^5}
$
where $\left\lfloor x \right\rceil$ denotes the nearest integer function.
and I ...
- 2,854
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How can I deal with floor functions inside a sum?
For context, imagine a sum along the variable $n$ whose terms $a_n = \lfloor nr\rfloor \cdot f(n)$ or $a_n = g(\lfloor nr\rfloor)$ for some value $r \in \mathbb{R}$. For example, I am working with the ...
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Reference request for floor sum identity $\sum\lfloor kn/m\rfloor+\sum\lfloor km/n\rfloor=(m-1)(n-1)/4$ (often linked to Landau or Eisenstein?)
I am seeking reliable references for the following identity, which holds for positive odd integers $m, n$ such that $\gcd(m, n) = 1$:
$$ \sum_{k=1}^{\frac{m-1}{2}} \left\lfloor \frac{kn}{m} \right\...
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Solve $\lfloor{x^{2}}\rfloor+13\lfloor{x}\rfloor-17=0$ for $x\in \mathbb{R}$
Solve The Equation $$\lfloor{x^{2}}\rfloor+13\lfloor{x}\rfloor-17=0$$ for $x\in \mathbb{R}$
My Thoughts
It is Given that $$\lfloor{x^{2}}\rfloor+13\lfloor{x}\rfloor-17=0$$
$$\implies 13\lfloor{x}\...
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