Used for questions and equations involving the floor function, which is defined to be the function that returns the largest integer less than or equal to $x$ (often denoted by $\lfloor x\rfloor$). See also (ceiling-function).

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-3
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0answers
33 views

Floor Function of Division of Factorials [on hold]

Find the value of $$\left\lfloor\frac{2002!}{2001! + 2000! + 1999! +\ldots+ 1!}\right\rfloor\;.$$ Please give a complete solution. Thanks!
4
votes
3answers
117 views

What is $\lfloor i\rfloor$?

So, floor is a function that converts a real number to an integer. It rounds down. This makes sense; however, what about complex numbers? I know that depending on the number, it can be split linearly. ...
1
vote
1answer
28 views

Inequality with floored squareroots

$(\lfloor \sqrt{n}\rfloor +1)^2\ge n+1$, for all $n\in \mathbb{N}$. I have convinced myself that this is true, but would like to see a formal proof.
1
vote
1answer
53 views

Area enclosed by the curve $\lfloor |x''| \rfloor +\lfloor |y''| \rfloor = 2$

The area enclosed by the curve $$\bigg\lfloor \frac{|x-1|}{|y-1|}\bigg\rfloor +\bigg\lfloor \frac{|y-1|}{|x-1|}\bigg\rfloor = 2\;,$$ Where $-2 \leq x,y\leq 0$ $\bf{My\; Try::}$ Let $x-1=x'$ and ...
1
vote
2answers
39 views

When does $\lfloor\sqrt{x^2 + n}\rfloor = \lfloor\sqrt{(x-1)^2 + n}\rfloor$?

Call $x \in[\sqrt{n},n/2] \cap \Bbb{Z} $ a critical point if the following holds $$\lfloor\sqrt{x^2 + n}\rfloor = \lfloor\sqrt{(x-1)^2 + n}\rfloor$$ It appears that mostly this does not happen. For ...
1
vote
1answer
38 views

Solve $\lfloor x \rfloor = ax+1$ for integral $a$

Solve $\lfloor x \rfloor = ax+1$, where $a$ is an integer. I have found the values of $x$ for $a=0$, $a=1$ and $a=-1$. But I don't know how to continue. How can I find the solutions for integral ...
1
vote
2answers
37 views

Summation over a floor function of a first degree polynomial

I've been trying to solve a difficult programming question for the last four days. I've gotten most of it done, but the piece I can't seem to figure out is this: Find a closed form expression of ...
4
votes
3answers
46 views

Sum of number-of-divisors function equals $\sum_{j=1}^{n} \lfloor n/j \rfloor$.

I am trying to prove the identity $$t(1) + t(2) + \cdots + t(n) = \Big\lfloor \dfrac{n}{1} \Big\rfloor + \Big\lfloor \dfrac{n}{2} \Big\rfloor + \cdots + \Big\lfloor \dfrac{n}{n} \Big\rfloor,$$ where ...
1
vote
1answer
16 views

Proof of $\sum_{j=1}^{p-1} \lfloor jq/p \rfloor = \frac{1}{2}(q-1)(p-1)$ Involving Pairing of Summands

I've seen the proof of the identity $$\sum_{j=1}^{p-1} \lfloor jq/p \rfloor = \frac{1}{2}(q-1)(p-1)$$ where $p$ and $q$ are coprime positive integers. This involves counting the remainders ...
0
votes
1answer
14 views

Moving terms outside of a floor

If I have $x^{\lfloor\frac{c}{a-b}\rfloor}$ is this equivalent to $(x^{\lfloor\frac{1}{a-b}\rfloor})^{\lfloor c \rfloor}$ if a,b,c are all integers and x is between 0 and 1? I'm concerned with this ...
2
votes
0answers
18 views

How to prove $|n-m|\lt d\implies |f(n)-f(m)|\lt e$ for the following condition?

How to prove $\exists n\in\Bbb{N},\forall e\in \Bbb{R^+},\exists d\in\Bbb{R^+},\forall m\in\Bbb{N}, |n-m|\lt d\implies |f(n)-f(m)|\lt e$ if $f(x)=\lfloor \frac{n}{3}\rfloor$. I don't know how to ...
1
vote
5answers
56 views

How to prove or disprove $\forall x\in\Bbb{R}, \forall n\in\Bbb{N},n\gt 0\implies \lfloor\frac{\lfloor x\rfloor}{n}\rfloor=\lfloor\frac{x}{n}\rfloor$.

How to prove or disprove $\forall x\in\Bbb{R}, \forall n\in\Bbb{N},n\gt 0\implies \left\lfloor\frac{\lfloor x\rfloor}{n}\right\rfloor=\left\lfloor\frac{x}{n}\right\rfloor$. So we want to prove ...
1
vote
1answer
74 views

Different ways of evaluating $n!$?

I've recently managed to prove the following result and was hoping to know if it already exists? $ \def\lf{\left\lfloor} \def\rf{\right\rfloor}$ $$ \ln(n!) = \sum_{k=1}^{p_k < n}\left( ...
2
votes
0answers
69 views

Find polynomial congruent to function modulo 6

How to find polynomial with rational coefficients, which is congruent to given function (or prove that it doesn't exist)? Particularly, how to find $P(x)$, so $$P(x)\equiv ...
1
vote
1answer
51 views

Sums involving floor function

I am looking for a direct formula for this sum $$\sum_{k=0}^n \lfloor{\sqrt{n+k}}\rfloor\lfloor{\sqrt{k}}\rfloor$$ Or a method to efficiently compute the sum for large n
1
vote
0answers
28 views

Floor Function and Euler-Mascheroni Constant

I need to know how to express the following function, $\lfloor{\sqrt{c^2-707}}\rfloor^2=y$ $(s.t.$ $c $ and $y$ are in $N)$, analytically. I think ...
1
vote
1answer
40 views

Number of values of $x\in [0,\pi]$ where $f(x)=\lfloor 4\sin x-7\rfloor$ is non derivable is?

Number of values of $x\in [0,\pi]$ where $f(x)=\lfloor 4\sin x-7\rfloor$ is non derivable is? $f(x)=\lfloor 4\sin x-7\rfloor=\lfloor 4\sin x\rfloor-7$ I drew the graph of the $f(x)$ and see that ...
1
vote
1answer
42 views

Prove $\bigg\lfloor\frac{\lfloor x \rfloor }{m}\bigg\rfloor =\bigg\lfloor\frac{x }{m}\bigg\rfloor $

Prove $\bigg\lfloor\frac{\lfloor x \rfloor }{m}\bigg\rfloor =\bigg\lfloor\frac{x }{m}\bigg\rfloor $ where $x\in \mathbb R , x\geqslant 0$ and $m\in \mathbb N$ What I did: Two cases: $x\in ...
2
votes
1answer
27 views

how to write floor function vectors in polar coordinates

let $$\lfloor{x}\rfloor=y$$ And $$z=x-\lfloor{x}\rfloor$$ Plot the following vector in polar coordinates: $$x\hat{\imath}+(y/z)\hat{\jmath}$$ I know that while transforming from cartesian to polar we ...
1
vote
1answer
27 views

Proof involving Big O and floor

Trying to prove or disprove this (pretty sure it's correct): Let $\mathcal{F}=\{f\mid f:\mathbb{N}\to\mathbb{R}^+\}$ $$\forall f\in\mathcal{F}: \left\lfloor \sqrt{\lfloor f(n)\rfloor }\right\rfloor ...
0
votes
1answer
26 views

Geometric series and floor function

I came out with a geometric sum with a floor function in the upper limit. It looks somehow like this: $$ \sum_{i=0}^{\lfloor 1/p \rfloor} (1-p)^i,\ \ 0<p<1.$$ I would like to study some ...
1
vote
1answer
57 views

Is there a simple way of proving that $\lfloor\sqrt{n}\rfloor+\lfloor\sqrt{4n+1}\rfloor = \lfloor\frac{3}{2} \lfloor \sqrt{4n+1} \rfloor\rfloor$?

It appears that $$\left\lfloor\sqrt{n}\right\rfloor+\left\lfloor\sqrt{4n+1}\right\rfloor = \left\lfloor\frac{3}{2} \left\lfloor \sqrt{4n+1} \right\rfloor\right\rfloor$$ for all $n \in \mathbb{N}_{0}$, ...
1
vote
0answers
49 views

Closed form of sum of rounded number

Let, there is two variable $N$ and $D$. Now, I want to find the closed form of the following sum: $$S_n = \sum_{k=1}^N \lfloor{\frac{k}{D}+\frac{1}{2}}\rfloor$$ Its easy to find the closed form by ...
1
vote
1answer
185 views

Proof that these two expressions are equivalent

I'm looking for algebraic proof that $$ y=\lim_{N \to \infty}\frac{\prod\limits_{n=0}^{N}x-n}{\prod\limits_{n=0}^{N-\lfloor x\rfloor}{x-\lfloor ...
5
votes
2answers
153 views

Is true that $\sum_{k=1}^n\frac{[kx]}{k}\leq[nx]$, for every $x\in\mathbb{R}$ and for every $n\in\mathbb{Z}^+$?

Is true that $\displaystyle\sum_{k=1}^n\dfrac{[kx]}{k}\leq[nx]$, for every $x\in\mathbb{R}$ and for every $n\in\mathbb{Z}^+$? My work: Let $x\in\mathbb{R}$ be given. For every $k=1,\dots,n$, define ...
7
votes
1answer
80 views

Is there a non brute force way to solve this problem?

A friend of mine asked me to prove or disprove that: $$ \left\lceil\frac{2}{2^{1/n}-1}\right\rceil=\left\lfloor\frac{2n}{\ln 2}\right\rfloor \forall n \in \mathbb{Z^+} $$ First of all, I run a ...
0
votes
1answer
33 views

Justification of a floor function simplification

Consider the expression: $\left\lfloor\frac{k+1}{2}\right\rfloor$, Where $k \in \mathbb{N}$ (natural numbers) How would I show that $\left\lfloor\frac{k+1}{2}\right\rfloor$ is the same thing as ...
-2
votes
2answers
39 views

Proving that $\lfloor x + k \rfloor=\lfloor x\rfloor + k$ [duplicate]

How to prove that $\lfloor x + k \rfloor=\lfloor x\rfloor + k$ when $k$ is an integer?
3
votes
2answers
31 views

If $a\in R$ and the equation $-3(x-\lfloor x \rfloor)^2+2(x-\lfloor x \rfloor)+a^2=0$ has no integral solution,then all possible values of $a$

If $a\in R$ and the equation $-3(x-\lfloor x \rfloor)^2+2(x-\lfloor x \rfloor)+a^2=0$ has no integral solution,then all possible values of $a$ lie in the interval $(A)(-1,0)\cup(0,1)$ $(B)(1,2)$ ...
0
votes
1answer
39 views

Help with equation that uses floor and ceiling functions

I have this equation and I've been stuck with it for a couple of hours. $\lfloor\log_2x\rfloor + 1 = \lceil\log_2(x+1)\rceil$ I've tried using this ceiling property: $\lceil x\rceil = n ...
0
votes
1answer
39 views

What does this floor expression evaluate to?

Consider the expression $$\left\lfloor\frac{2n-1}{2}\right\rfloor\;:$$ does it make sense that this floor function will evaluate to $n$? Or should it be $n-1$?
1
vote
1answer
57 views

Proof with induction on a sequence

Let $\{a_k\}_{k=0}^\infty$ be a sequence where $a_0 = 0$ $a_1 = 0$ $a_2 = 2$ $\forall k \geq 3, a_k = a_{\lfloor k/2 \rfloor} + 2$ Show that every element of this sequence is even. I am ...
-7
votes
1answer
69 views

Using Induction to prove a sequence [closed]

How do i prove that $a_{n+1}$ is even? I have tried many things but can't seem to get anywhere in turns of proving. Define the sequence $(a_n)$ as follows $a_0 = 0$, $a_1 = 0$, $a_2 = 2$, $a_n = ...
0
votes
1answer
30 views

What is meant by $[x]$ in this case?

I'm reading a report on the $\space$Gamma function$\space$ and in the first segment it talks about where the following representation of the Gamma function converges: $$\Gamma(x)= \int^{\infty}_{0} ...
2
votes
1answer
87 views

Function that produces sequence 112123123412345…

I'm trying to find a function/formula for $a_n$ such that it produces the sequence $112123123412345$ and so on. I know that one possible way to do this is to find a function like $n-b_n$ where $b_n$ ...
2
votes
1answer
86 views

Iterating through the integerial points of $f(x)$

The floor function $g(x)=\left\lfloor f(x) \right\rfloor$ jumps at points where $f(x)$ is an integer. What I want is a function that gives the $n$-th jump point from $g(0)$. So, for let's say ...
0
votes
2answers
57 views

How to isolate a variable in a floor function?

I hope someone may be able to help me solve the following equation for $y$? $$x=\frac b{50}\left\lfloor{50ay\over b}\right\rfloor$$ I'm trying to isolate $y$ so I can program an Excel file to solve ...
2
votes
2answers
61 views

If $x$, $\{x\}$, $\lfloor x\rfloor$ are in G.P, find $x$.

If $x$, $\{x\}$, $\lfloor x\rfloor$ are in Geometric Progression, find $x$; $x \neq 0$. Here, $\{x\}=x-\lfloor x\rfloor$ Some properties are pretty evident: $$0\leq \{x\} < 1 \tag{1}$$ ...
0
votes
1answer
91 views

For all real numbers $x$, prove $\lfloor x - 2\rfloor = \lfloor x\rfloor - 2$ [closed]

Prove the following statement: For all real numbers $x$, $\lfloor x - 2\rfloor = \lfloor x\rfloor - 2$ I'd appreciate some help with this. All I know is that the floor function $n$ implies : $n ...
0
votes
2answers
51 views

Function with an asymptote at y=-1 and y=1

I'm looking for a function that has two asymptotes parallel to the x-axis. Preferably it should also only cross the x-axis at (0,0) and be built without using any trigonometric functions. Mind you, if ...
0
votes
1answer
37 views

What floor function identity makes this true?

I know that the graph of these two functions is the same: $$(-1)^{\lfloor x\rfloor} = -2\lfloor x\rfloor + 4\left\lfloor\frac {\lfloor x\rfloor}2\right\rfloor + 1$$ Both of them interchange sign at ...
-1
votes
3answers
60 views

How to evaluate the finite sequence (involving the floor function)? [closed]

How to sum this:$$ \lfloor\sqrt{1}\rfloor + \lfloor\sqrt{2}\rfloor + \lfloor\sqrt{3}\rfloor +\lfloor\sqrt{4}\rfloor + \ldots + \lfloor\sqrt{50}\rfloor = ?$$
11
votes
1answer
584 views

odd prime division

Prove that if $p$ is an odd prime then $p$ divides $\lfloor(2+\sqrt5)^p\rfloor -2^{p+1}$ I am struggling to progress with this question. Here is my working out so far: Page 1 working out Page 2 ...
4
votes
1answer
24 views

When Is the Remainder Less Than The Quotient

I was given the problem: Let $f(n)$ be the number of times $a$ is $n$-well for $1\le a \le n$. An integer, $a$, is $n$-well if $$\left\lfloor\frac{n}{\left\lfloor ...
1
vote
5answers
48 views

Floor-Ceil Properties

If $n$ and $k$ are integers, with $k$ different from zero: $$\left\lceil{\frac{n+1}{k}}\right\rceil = \left\lfloor{\frac{n}{k}}\right\rfloor + 1$$ How can I prove this property? I would appreciate ...
2
votes
1answer
61 views

What is the value of $\lim _{x\to 0}\left\lfloor\frac{\tan x \sin x}{x^2}\right\rfloor$

How do I evaluate the limit $$\lim _{x\to 0}\left\lfloor\frac{\tan x \sin x}{x^2}\right\rfloor$$ where $\lfloor\cdot\rfloor$ denotes greatest integer function. I know that $x>\sin x$ and $x < ...
2
votes
2answers
102 views

Proving $\lim_{n\to\infty}\sum_{k=n}^{\infty}\Big(\frac{1}{n\left\lfloor\frac kn\right\rfloor}-\frac1k\Big)=\gamma$

I remember arriving at the following equality: $$\lim_{n\to\infty}\sum_{k=n}^{\infty}\left(\frac{1}{n\left\lfloor\frac kn\right\rfloor}-\frac1k\right)=\gamma$$ where $\gamma$ denotes the ...
0
votes
1answer
41 views

Showing that percentage of the coefficients of $(x+y)^n$ being even tends towards $100\%$ when taking the limit of $n$ to infinity

A couple of weeks ago I came up with this function which could determine the number of coefficients divisible by some number $m$ in the binomial expansion of the expression $(x+y)^q$: iff ...
2
votes
4answers
89 views

For all $x$ which are real numbers, prove that $\lfloor 2x\rfloor = \lfloor x\rfloor + \lfloor x+0.5\rfloor.$ [duplicate]

For all $x$ which are real numbers, prove that $$\lfloor 2x\rfloor = \lfloor x\rfloor + \lfloor x+0.5\rfloor.$$ I know that Let $\lfloor x\rfloor = n$ $n \leq x < n+1$
4
votes
4answers
66 views

Inequalities with floor function

I need some help with this exercise, I'm pretty new solving this exercises. $$ \lfloor x \rfloor + \lfloor y \rfloor \le \lfloor x + y \rfloor \le \lfloor x \rfloor + \lfloor y \rfloor + 1$$ I know ...