Skip to main content

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).

Filter by
Sorted by
Tagged with
1 vote
1 answer
59 views

A conceptual misunderstanding in limits and geometric series

So, it starts off like this: My friends told me that the floor function of 0.9 repeating (a.k.a. 0.99999.... ) is 0, which is factually untrue since 0.9 repeating is known and proven to be exactly ...
32 Bit's user avatar
  • 13
0 votes
0 answers
12 views

Clarification on Theorem 6 from Uniswap V1 Formalized Model: $e_0 < e_2$ and $t_0 < t_2$

I have been studying the formalized model of Uniswap V1 from the whitepaper and the accompanying formalized model document (left the links below in the references). Specifically, I am focusing on ...
Vegetal605's user avatar
  • 1,171
0 votes
1 answer
96 views

Floor function with factorials

$x$ is an integer satisfying: $$\left[\frac x{1!}\right]+\left[\frac x{2!}\right]+\left[\frac x{3!}\right] + \cdots +\left[\frac x{10!}\right]=1001.$$ Find largest prime divisor of $x$, where $[~]$ ...
Quark's user avatar
  • 3
0 votes
0 answers
44 views

$f(x)=\frac{x\lfloor x\rfloor}{1+x^2}$. Find range.

Find range of the function $f(x)=\frac{x\lfloor x\rfloor}{1+x^2}$. My Attempt: I found this problem really weird. On plotting the graph i observed that $f(x)$ is not able to take value between $0.5$ ...
Maverick's user avatar
  • 9,471
0 votes
1 answer
33 views

Solving an equation involving Floor and Ceiling

Trying to see if there is a mathematical expression to represent Floor and Ceilings within an equation and hence subsequently solve it. The one in question is: $⌊x⌋*⌈x⌉=x^2$ where 0<x<=100,...
Steve237's user avatar
  • 187
0 votes
0 answers
45 views

⌊a/2⌋+⌊a/3⌋+⌊a/5⌋ = a [duplicate]

I found this problem in my elementary number theory textbook. find all the integers 'a' such that ⌊a/2⌋+⌊a/3⌋+⌊a/5⌋ = a I did try viewing each of ⌊a/2⌋, ⌊a/3⌋, ⌊a/5⌋ as the number of multiple of 2, 3, ...
jihu song's user avatar
3 votes
2 answers
93 views

How to show $\lfloor\frac{n}{2}\rfloor\lfloor\frac{n-1}{2}\rfloor\lfloor\frac{m}{2}\rfloor\lfloor\frac{m-1}{2}\rfloor$ is equal to this expression?

I am trying to show equality between the two expressions $$\frac {\lfloor{\frac n 2}\rfloor(\lfloor{\frac n 2}\rfloor - 1)\lfloor{\frac m 2}\rfloor(\lfloor{\frac m 2}\rfloor - 1)} 4 + \frac {\lfloor{\...
Princess Mia's user avatar
  • 2,493
0 votes
0 answers
49 views

Limit of an Integral with a Floor Function

I'm looking to evaluate the following integral and would like to verify my answer: $$\lim_{n\to\infty} \frac{1}{n^2} \int_{0}^{n^2} \sqrt{\frac{n^2-\lfloor \sqrt{x}\rfloor^2}{x}} \ \textrm{d}x$$ I've ...
Stamp's user avatar
  • 399
1 vote
2 answers
77 views

Confused on answer given for the question: prove $\lceil\frac{n}{m}\rceil = \lfloor\frac{n+m-1}{m}\rfloor$ [closed]

Here is the link to the problem and Mr. Andrew's solution: 1https://math.stackexchange.com/a/1281757/801877 Surprisingly, when trying to prove this problem I tried doing this exact way and got ...
Bob Marley's user avatar
0 votes
1 answer
52 views

Trying to see if $\lfloor x - y\rfloor = \lfloor x\rfloor - \lfloor y\rfloor$ [duplicate]

So let $\lfloor x\rfloor = n,\ \lfloor y\rfloor = m$ Then by definition we know that: $ n \leq x < n + 1$ and $ m \leq y < m + 1$ But when I subtract the 2 inequalities together, I get $n - m \...
Bob Marley's user avatar
1 vote
1 answer
62 views

Confusion on floor and ceiling function proof question.

These are questions from the Discrete Math textbook by Rosen, and I was just confused on what they're exactly asking: specifically when they write "... , when it is the larger/smaller of 2 ...
Bob Marley's user avatar
3 votes
0 answers
34 views

Does half of the sequence $\left(n^{\alpha}\right)_{n=1}^{\infty}$ have even integer part/floor?

Define $e\left((a_n)_{n=1}^{\infty};N\right)$ to be the amount of members of the sequence $(a_n)_{n=1}^{\infty}$ that are $\leq N$ and have even integer part (also known as the floor of the number). ...
Adam Rubinson's user avatar
0 votes
1 answer
50 views

The sum of Floor function

I am looking of calculating the following Sum. $\sum\limits_{k=0}^m \lfloor n/2^k \rfloor $ when $l$ is an integer and satisfies $n=2^m+l, 0\le l <2^m$ So far I’ve got $\sum\limits_{k=0}^m \lfloor ...
Russel0201's user avatar
2 votes
4 answers
113 views

$\left[\dfrac 12\displaystyle\sum_{n=1}^{k^2}\frac 1{\sqrt n}\right]=k-1$

Consider this $\left[\dfrac 12\displaystyle\sum_{n=1}^{k^2}\frac 1{\sqrt n}\right]$ where $[\cdot]$ is the greatest integer function. I had observed that its value is $(k-1)$ by putting $k=2,3,4,$ ...
user1318878's user avatar
4 votes
1 answer
116 views

Accumulation points of the sequence $c_n = \lfloor \cos(\sqrt{n}) \rfloor$

I'm trying to find the accumulation points for the sequence $c_n = \lfloor \cos(\sqrt{n}) \rfloor, n \in \mathbb{N}_0$. I know what the points are, but I'm having trouble coming up with an explicit ...
Focus's user avatar
  • 43
0 votes
0 answers
75 views

Looking for a solution of $\sum_{i = 1}^{k} \sum_{{d}_{1}\, {d}_{2} = i (2k - i), {d}_{1} \le N, {d}_{2} \le N} [GCD(2 k, {d}_{1}, {d}_{2}) = 1]$

The double sum is $$\sum_{i = 1}^{k} \sum_{\substack{{d}_{1}\, {d}_{2} = i \left({2k - i}\right), \\ {d}_{1} \le N, {d}_{2} \le N}} \left[{\left({2\, k, {d}_{1}, {d}_{2}}\right) = 1}\right]$$ where [.....
Lorenz H Menke's user avatar
3 votes
1 answer
224 views

Solve for $x$: $x\lfloor x\lfloor x\lfloor x\rfloor\rfloor\rfloor=2001$

Solve for $x$: $x\lfloor x\lfloor x\lfloor x\rfloor\rfloor\rfloor=2001$ Given solution: $\lfloor x\rfloor=\lfloor \sqrt[4]{2001}\rfloor=\{6,-7\}$ If $\lfloor x\rfloor=6\Rightarrow x\lfloor x\lfloor 6x\...
Maverick's user avatar
  • 9,471
2 votes
1 answer
97 views

Solve for $x$: $\lfloor 2^x\rfloor+\lfloor 3^x\rfloor=\lfloor 6^x\rfloor$

Solve for $x$: $\lfloor 2^x\rfloor+\lfloor 3^x\rfloor=\lfloor 6^x\rfloor$ My Attempt: Clearly the equation is satisfied for all $x<0$. For positive $x$, $2^x+3^x-6^x=\{2^x\}+\{3^x\}-\{6^x\}$ $-1<...
Maverick's user avatar
  • 9,471
11 votes
2 answers
358 views

What is $\int_{0}^{\pi/2}\frac{\operatorname{lcm}(a\cos x,a\sin x)}{a^2}dx$?

I came up with this while messing around with the $\gcd$ and $\operatorname{lcm}$ functions in Desmos. $$I(a)=\int_{0}^{\pi/2}\frac{\operatorname{lcm}(a\cos x,a\sin x)}{a^2}dx$$ The function inside ...
Dylan Levine's user avatar
  • 1,676
0 votes
0 answers
36 views

Challenge: Define rounding functions using integer checking functions

This question was inspired by Willans' formula, in which he used the cosine function as a way to constrain a number and floor it to be able to count primes (see https://www.youtube.com/watch?v=...
Yusuf Maher's user avatar
2 votes
0 answers
35 views

Finding the sum of the floor function of $a,(b-1)/2,c$ given two symmetric sums

Problem: Let $a<b<c$ be $3$ real numbers satisfying $a+b+c=6$, $ab+bc+ca=9$. Then, determine the value of $\lfloor{a}\rfloor+\lfloor\frac{b-1}{2}\rfloor+\lfloor{c}\rfloor$. My method of solution:...
Cognoscenti's user avatar
2 votes
3 answers
89 views

Calculate the whole part of $A=\frac{1012^2}{\sqrt{1*2}+\sqrt{3*4}+\cdots+\sqrt{2023*2024}}$

the question Calculate the whole part of: $$A=\frac{1012^2}{\sqrt{1*2}+\sqrt{3*4}+\cdots+\sqrt{2023*2024}}$$ my idea My first thought was to to fit the number between 2 consecutive natural numbers ...
IONELA BUCIU's user avatar
  • 1,115
1 vote
0 answers
29 views

Looking for a solution to a 2-dimensional recurrence relation with a floor function in GF(2)

I have a recurrence relation that i'd like to solve. It is defined as follows: $$ a_{i,j} = (a_{i-1,j} + \lfloor 3 \sum_{k=0}^{j-1} \frac{a_{i-1,k}}{2^{j-k}} \rfloor \space) \bmod 2 $$ where $ a_{i,...
BlueLisztmaths's user avatar
0 votes
1 answer
17 views

Y~Geometric(p) then why does the CDF of Y have the floor function?

Here is the screenshot from my textbook, Hossein Pishro-Nik Here, I have calculated the CDF of Y to be 1 - q^y. But the next step involved y being a floor function. I am not sure why that is the case ...
Arnav Aditya's user avatar
4 votes
0 answers
156 views

Positive solutions to a linear Diophantine equation

Let $d,d',n\in \mathbb N$ be given. If you want, assume $(d,d')=1$. How many positive integer solutions does $$dx+d'x'=n$$ have? (Assuming $(d,d')=1$). I know there are $n/dd'+\mathcal O(1)$ solutions,...
tomos's user avatar
  • 1,662
0 votes
0 answers
65 views

Weird Property of Irrational Numbers and their "Conjugate"

Let us have a irrational number $a$. Let $a'$ be $\frac{1}{a'} + \frac{1}{a} = 1$. If we let $A$ and $B$ be sets such that $A =$ {$\lfloor a \rfloor, \lfloor 2a \rfloor, \lfloor 3a \rfloor, \lfloor 4a ...
bob john's user avatar
0 votes
0 answers
29 views

Proving the Greatest Integer Fucntion (G.I.F) of (50!/e + 1/2) is D50(derangement)

Prove that, If D(n) denotes dearrangement for n objects, then D(50) = [ 50/e! + 1/2 ] Where [.] denotes the greatest integer function. Through exapnsion i generated 50/e! but how can we adjust the &...
OpateItZOpatoOpate's user avatar
0 votes
0 answers
20 views

Points of discontinuity of greatest Integer function

So here's my problem Find the points of discontinuity of $[x^2]$ in the interval$[-1,2]$ where [] is the G.I.F. I am not able to understand why at $2^{1/2}$ and $3^{1/2}$ the function is being ...
πααρτθ Σαρθι's user avatar
1 vote
1 answer
21 views

Quesiton regarding nature of equations involving fractional part and greatest integer functions

This question is in context of the following problem Solve: $[x]^2 = x + 2\{x\}$ Where $[.]$ and $\{.\}$ represent the greatest integer and fractional part function respectively. The solution for the ...
koiboi's user avatar
  • 736
3 votes
2 answers
35 views

Question regarding intersection of values of a function

This question is being asked in regards to the following problem Solve: $x^2 - 4 - \lfloor x \rfloor = 0$ The solution for the above equation in my textbook is done by drawing the graph of both the ...
koiboi's user avatar
  • 736
1 vote
2 answers
38 views

Question regarding property of greatest integer function

In my textbooks section about the properties of the greatest integer function the two following properties looked quite interesting and i could not understand them right away. Property 1 $$\lfloor(\...
koiboi's user avatar
  • 736
0 votes
1 answer
61 views

Computing $h(h(x))$ where $h (x) = \lfloor 5x - 2 \rfloor$

In Velleman's "Calculus: a Rigorous Course," Example 9 from Section 1.3 tasks us with computing $ h(h(x)) $, where $ h(x) = \lfloor 5x - 2 \rfloor $. My initial solution: \begin{align*} h(\...
F. Zer's user avatar
  • 2,357
1 vote
2 answers
68 views

How to compute the following limit? $\lim\limits_{x\to\ 0^+} {\frac{x-\lfloor x \rfloor}{x+\lfloor x \rfloor}}$ [closed]

$$\lim\limits_{x\to\ 0^+} {\frac{x-\lfloor x \rfloor}{x+\lfloor x \rfloor}}$$ Here, $\lfloor x \rfloor$ represents the floor of $x$. I tried using a graphing calculator (desmos) to plot the function $...
Jesko's user avatar
  • 45
4 votes
1 answer
164 views

Prove that sum of integrals $= n$ for argument $n \in \mathbb{N}_{>1}$

ORIGINAL QUESTION (UPDATED): I have a function $f:\mathbb{R} \rightarrow \mathbb{R}$ containing an integral that involves the floor function: $$f(x):= - \lfloor x \rfloor \int_1^x \lfloor t \rfloor x \...
Richard Burke-Ward's user avatar
2 votes
3 answers
259 views

Prove⌈a/b⌉ ≤ a/b + (b-1)/b [closed]

For integers $a, b > 0$, Prove $⌈a/b⌉ ≤ (a + (b-1))/b$ RHS $= a/b + (b-1)/b $ where $ (b-1)/b $ is $[0,1)$ If $a/b$ is an integer, inequality holds true as we are adding non-negative term. If $a/b$ ...
jam's user avatar
  • 81
0 votes
1 answer
83 views

For $0<x<1$, let $f(x)=\int_0^1\left( \left\lfloor\frac{x}{y}\right\rfloor-x \left\lfloor\frac{1}{y}\right\rfloor\right)dy\ldots$

For $0<x<1$, let $f(x)=\int_0^1\left( \left\lfloor\frac{x}{y}\right\rfloor-x \left\lfloor\frac{1}{y}\right\rfloor\right)dy$, $\lfloor. \rfloor$ denotes the greatest integer function. If $f(x)$ ...
mathophile's user avatar
  • 3,815
2 votes
1 answer
34 views

Asymptotics of expression involving floor functions

Consider two functions $f_1(n)$ and $f_2(n)$ that grow to infinity at the same speed, say $$\lim_{n\to \infty} \frac{f_1(n)}{f_2(n)}=c $$ for some $c>0$. I am studying the squared difference of $...
FishPie's user avatar
  • 43
1 vote
2 answers
100 views

$f(n)= \left\lfloor \frac{an+b}{cn+d} \right\rfloor , \forall n \in \mathbb N$ is surjective

Let $a,b,c,d \in \mathbb N , d \ne 0$ and consider the funtion $f: \mathbb N \rightarrow \mathbb N $ such that : $$f(n)=\left \lfloor \frac{an+b}{cn+d}\right \rfloor , \forall n \in \mathbb N$$ Prove ...
Unknowduck's user avatar
2 votes
4 answers
125 views

Graphing $(\lfloor x \rfloor + \lfloor1-x\rfloor)$

$\lim_{x\to0+}(\lfloor x \rfloor + \lfloor1-x\rfloor)$ $\lim_{x\to0-}(\lfloor x \rfloor + \lfloor1-x\rfloor)$ I tried to solve by graphing $(\lfloor x \rfloor + \lfloor1-x\rfloor)$ Graph of the ...
Ak9848's user avatar
  • 35
2 votes
3 answers
97 views

Solving equation with two floor functions

I'm trying to solve the following question \begin{equation*} \left\lfloor \frac{\left\lfloor \frac{3\lfloor x\rfloor }{2}\right\rfloor }{9}\right\rfloor =4 \end{equation*} I got the following ...
dagoda's user avatar
  • 31
2 votes
0 answers
33 views

Solve recurrence $A(0)=0$, $A(n)=1+\min_{1\leq k\leq n}\left\{\max\left(\left\lfloor\frac{k-1}{2}\right\rfloor, A(n-k)\right)\right\}$.

I am asked to solve the following recurrence problem: $$A(0)=0,\quad A(n)=1+\min_{1\leq k\leq n}\left\{\max\left(\left\lfloor\frac{k-1}{2}\right\rfloor, A(n-k)\right)\right\}\,. $$ I have observed ...
YanRuiJie's user avatar
0 votes
1 answer
47 views

integration of floor function [closed]

How to Integrate the following? $$\int_{2}^{343}\{x-\lfloor{x}\rfloor\}^2dx = \int_{2}^{3}x^2dx + \int_{3}^{4}x^2dx+\cdots+\int_{342}^{343}x^2dx$$ This is what I did, How can I proceed further with ...
econhead's user avatar
0 votes
0 answers
18 views

How to reduce division with uprounding to integer-arithmetic operations with downrounding if the denominator is not necessarily an integer?

Let 𝑖∈ℕ₀ be an unknown variable and 𝑐∈ℚ₊ be a known constant such that 𝑐≥2 and 100𝑐 ∈ ℕ₊. (So we can pre-compute anything regarding 𝑐, e.g., represent 100𝑐 as a product of powers of primes.) We ...
AlMa1r's user avatar
  • 101
3 votes
5 answers
114 views

Solve $\lfloor x-1\rfloor(3^x-2^x-\lfloor x^2\rfloor) = 0$

Solve the following equation in $\mathbb R$ $$\lfloor x-1\rfloor(3^x-2^x-\lfloor x^2\rfloor) = 0$$ where $\lfloor y\rfloor=k \iff k \le y < k+1 , k \in \mathbb Z$ I will address 2 cases. $\lfloor ...
Unknowduck's user avatar
0 votes
0 answers
46 views

Inverse of a piecewise floor function

Given the following definitions $$ y(x) = \left\lfloor{ (x - n + a - b)/a }\right\rfloor ,\quad a,b \in \mathbb{R}, n \in \mathbb{N}, $$ $$ z(x) = \max (0, y(x)), $$ where $x, y(x), z(x) \in \mathbb{N}...
Toool's user avatar
  • 307
1 vote
2 answers
78 views

Simplifying $\sum _{t=1}^{\infty }\:\frac{\left(1+g\right)^{ \lceil t/2\rceil}}{\left(1+k\right)^t}$

how to simplify this expression so that I get an expression only? when $t = 1, 2,.... $ $$y = \sum _{t=1}^{\infty }\:\frac{\left(1+g\right)^{ \lceil t/2\rceil}}{\left(1+k\right)^t}$$ I know that for t ...
CountDOOKU's user avatar
  • 1,041
0 votes
2 answers
46 views

Inequality for integers with floor function

I want to show that for any nonnegative integers $l$ and $b$ we have $$ \frac{l}{2^{b+1}} - 1 \leq \left\lfloor \frac{l-1}{2^{b+1}} \right\rfloor. $$ I have a proof where I wrote $l = \alpha\cdot 2^{...
Lereu's user avatar
  • 434
2 votes
3 answers
73 views

Solve in $\mathbb{R}^+$ algebra calculus Equation with derivatives

I was trying to solve this question: Solve in $\mathbb{R}^+$ the Equation $7^x-5^x=\lfloor x^2 \rfloor+1$ Where $\lfloor t\rfloor$ is the floor function It is known that $\lfloor x^2\rfloor\leq x^2 $ ...
Kolsinki Vetko's user avatar
0 votes
1 answer
41 views

Prove that the maximum difference between $\frac{n}{6}$ and $\lfloor \frac{(n-3)}{6} \rfloor$ is $1 \frac{1}{3}$?

Prove that the maximum difference between $\frac{n}{6}$ and $\lfloor \frac{(n-3)}{6} \rfloor$ where $n$ is an integer is $1 \frac{1}{3}$? I know this is true because for $n = 8 + 6i$ where $i$ is an ...
Kenneth Watanabe's user avatar
1 vote
1 answer
93 views

Is there a nice form to $\sum_{k=1}^{n}\left\lfloor\sqrt{ r^2-k^2 }\right\rfloor$?

$$\sum_{k=1}^{n}\left\lfloor\sqrt{ r^2-k^2 }\right\rfloor$$ where $r$ is a constant (not necessarily an integer). Note that $r\ge n$ and both $r, n$ are positive. I apologize if I'm not able to ...
Lucien Jaccon's user avatar

1
2 3 4 5
46