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).

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44 views

Integral involving a floor function

I've been thinking about this problem for a bit: $$\lim_{n \to \infty} \frac{1}{n} \int_1^n \log x \left( \left\lfloor \frac{n}{x-1} \right\rfloor- \sum_{k=1}^\infty \left\lfloor \frac{n}{x^k} \right\...
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1answer
24 views

Efficient Meromorphic Approximation For Getting the ith Bit of a Number

Thanks to this answer, I know that to get the $i$th bit of a number $n$, you can do $$\left\lfloor\frac{n}{2^i}\right\rfloor-2\left\lfloor\frac{n}{2^{i+1}}\right\rfloor$$ However, I need this formula ...
5
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2answers
74 views

Find the greatest integer less than $\frac{1}{\sin^2(\sin(1))}$ without calculator.

Find the greatest integer less than $$\frac{1}{\sin^2(\sin(1))}$$ This was on one of my tests. All angles in radians. Here's my work: $$0<1<\frac{\pi}{3}<\frac{\pi}{2}$$ Since $\sin(x)$ is ...
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1answer
38 views

Find an approximation to $\sum _{j=1}^x \left\lfloor \frac{x}{j} -1\right\rfloor (j-1)$

I want to find an approximate (ideally asymptotic) function $f_1:\mathbb{R}\to\mathbb{R}$ in order to approximate a function $f_0:\mathbb{R}\to\mathbb{N}$ with $f_0$ defined by $$\sum _{j=1}^x \left\...
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2answers
52 views

Find the smallest positive real $x$ such that $\lfloor{x^2}\rfloor - x\lfloor{x}\rfloor = 6$

Problem Find the smallest positive real $x$ such that $\lfloor{x^2}\rfloor - x\lfloor{x}\rfloor = 6$. What I've done I set $m = \lfloor x \rfloor$ and $n = \{x\}$. Then I proceeded as below: $\lfloor (...
3
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1answer
87 views

Prove an Elementary sum of floor function

Prove: If $a$ and $b$ are odd and relatively prime, $$\sum_{\substack{0 \lt x \lt b/2\\x \in Z}} \left\lfloor \frac{ax}{b} \right\rfloor + \sum_{\substack{0 \lt y \lt a/2\\y\in Z}} \left\lfloor \frac{...
2
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1answer
69 views

What is the range of $x,y,z$ when $n$ is a known natural number in: $n=x^5+y^5+z^5$

I have the following question: What is the range of the sum of three distinct natural numbers to the fifth power than are equal to a known natural number? Mathematically speaking: $$n=x^5+y^5+z^5\...
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0answers
58 views

Prove that $\sum_{r=2}^{n} \left \lfloor n^{\frac{1}{r}} \right \rfloor = \sum_{r=2}^{n} \left \lfloor \log_{r}(n) \right \rfloor$.

Prove that $$\sum_{r=2}^{n} \left \lfloor n^{\frac{1}{r}} \right \rfloor = \sum_{r=2}^{n} \left \lfloor \log_{r}(n) \right \rfloor\,.$$ I have tried to use substitutions of $n=p^k$ in order to try ...
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3answers
205 views

The number of ways to represent a natural number as the sum of three different natural numbers

Prove that the number of ways to represent a natural number $n$ as the sum of three different natural numbers is equal to $$\left[\frac{n^2-6n+12}{12}\right].$$ It was in our meeting a year ago, but I ...
9
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1answer
139 views

Solve $\lfloor \ln x \rfloor \gt \ln \lfloor x\rfloor$

The question requires finding all real values of $x$ for which $$\lfloor \ln x\rfloor \gt \ln\lfloor x\rfloor $$ To start off, one could note that $$\lfloor \ln x \rfloor =\begin{cases} 0,& x\in[1,...
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28 views

Can closed forms for the outputs in this production simulation be found?

(I am not sure whether this question should better be asked on Game Development SE but I think of this more as a mathematical problem than one specific to game development.) For a game I am ...
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0answers
51 views

proof involving floor functions [duplicate]

I am looking at a problem that roughly says If $a,b \in [0,1)$, and $\lfloor na\rfloor+\lfloor nb\rfloor=\lfloor n(a+b)\rfloor$ for every $n\in\mathbb{N}$, then $a=0$ or $b=0$. The proof that was ...
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0answers
24 views

Inequalities related to ceil function

So I need to find a function say $g(x)$ that upper bounds another function say $f(x)$; On that same note suppose $f(x)$ is a ceil function I am acquainted with the inequality that holds for ceil ...
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3answers
61 views

Does $\int_0^{\pi \over 2} \lfloor \tan(x) \rfloor\, dx$ converge?

My book follows the following method Let $$I=\int_0^{\pi \over 2} \lfloor \tan(x) \rfloor\, dx.$$ Then using King's rule $$\int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-x)dx$$ we have $$I=\int_0^{\pi \...
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1answer
38 views

Motivation confusion in floor and ceiling function algebra

From HMMT https://hmmt-archive.s3.amazonaws.com/tournaments/2019/nov/team/solutions.pdf: Compute the sum of all positive real numbers $x \le 5$ satisfying $$x=\frac{\left\lceil{x^2}\right\rceil+\left\...
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2answers
149 views

$\lfloor\frac12+\frac1{2^2}+\frac1{2^3}+\cdots\rfloor\;$ vs $\;\lim_{n\to\infty}\lfloor\frac12+\frac1{2^2}+\cdots+\frac1{2^n}\rfloor$

Is there any difference between answers of $[1]$ and $[2]$? $$\Bigg\lfloor\frac12+\frac1{2^2}+\frac1{2^3}+\cdots\Bigg\rfloor \tag*{$\space.....[1]$}$$ $$ \lim _{n \rightarrow \infty} \Bigg\lfloor\frac{...
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0answers
21 views

Finding K depending upon the following criteria?

If $\sum_{n=1}^{k}\left[\frac{1}{3}+\frac{n}{90}\right]=21,$ where $[x]$ denotes the integral part of $x,$ then $k=$ (a) $84$ (b) $80$ (c) $85$ (d) none of these $21=\sum_{n=1}^{k}\left[\frac{1}{3}+\...
11
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1answer
205 views

Detailed analysis of the secretary problem

The secretary problem is well-known. $N>2$ candidates present themselves for a job. You must either hire a candidate immediately after interviewing him, or let him go. The strategy you follow is ...
1
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1answer
36 views

Floor function sequence [closed]

$\{a_n\}$ is defined as follows: $a_1 = 1, \ a_{n+1}=\lfloor (3+2\sqrt2)a_n\rfloor$ Show that $a_n$satisfies $a_{n+2}=6a_{n+1}-a_n$. I'd appreciate it if you could share your ideas.
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2answers
36 views

Simplification of $ [(n+1) \alpha ]- [ n \alpha]$

What can we say about the value of $ [(n+1) \alpha ]- [ n \alpha]$, where $ \alpha$ is any irrational number? Can this be further simplified? Here $[x]$ denotes the largest integer less or equal to $x$...
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1answer
68 views

Floor equation $x=\frac{⌊x⌋}{x - ⌊x⌋}$

$$x=\frac{⌊x⌋}{x - ⌊x⌋}$$ I've tried putting $x = n + e$, where $n\in\mathbb N, 0 \le e < 1$ and I've got that $n + e = \frac{n}{e}$. Now, I don't know how should I approach problem further, and I ...
2
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2answers
76 views

Floor function equation $⌊x + 1/2⌋ + ⌊x⌋ = \frac12 x^6$ [closed]

So in this floor equation $⌊x + 1/2⌋ + ⌊x⌋ = \frac12 x^6$, I've tried putting $x = n + e$, where $0 \le e < 1$, but I didn't get anything useful. What should be an approach in these situations?
2
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1answer
125 views

How to solve those two polynomials which contains a lot of floor function?

Solve for $x$ : $$3x^2-2x\lfloor x\rfloor + 4\lfloor x^2\rfloor + x - 4\lfloor x\rfloor-\frac{7}{2}=0$$ Then solve for $x$ : $$\lfloor 3x^2-2x\lfloor x\rfloor + 4\lfloor x^2\rfloor + x - 4\lfloor x\...
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1answer
20 views

Finding the number of possible values of $\lfloor x \lfloor x \rfloor \rfloor$ for $0 \le x \le 10.$

Find the number of possible values of $f(x) = \lfloor x \lfloor x \rfloor \rfloor$ for $0 \le x \le 10$. I tried to see if I could set up some inequality, or do some casework, but I still can't get ...
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1answer
33 views

How can we erase graph of $f(x)=-10a((x/a)-[x/a])$ from specific parts?

The function $$f(x)=-10a(\frac{x}{a}-[\frac{x}{a}])$$ I want to erase some parts of graph from $x=a$ to $2a$ and from 3a to 4a ... And so on How can i accomplish that? I have no idea of how can we do ...
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1answer
30 views

Is there any way to shift distance between zeroes of function $-10(x-[x])$ by $a$ units on x axis?

The function is $$-10(x-[x])$$ Now i wonder if we can shift distance between zeroes of this function by a units is it possible? I tried replacing x with x-1 but it doesn't work
1
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1answer
36 views

How to describe ceiling- and floor-like functions that round to a specific decimal place?

I am trying to describe floors and ceilings with non-integer factors. Rather than rounding up or down to the nearest integer, I need to for example round to the nearest 0.1. For example, in what I'm ...
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2answers
52 views

Efficiently computing $\sum_{\sqrt{N} \lt p \in primes \leq N} \lfloor \frac{N}{p} \rfloor$

I'm interested in computing $f(N) = \sum_{\sqrt{N} \lt p \in primes \leq N} \lfloor \frac{N}{p} \rfloor$ $N$ is large enough that primes up to $\sqrt{N}$ are available, but not much beyond and ...
0
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1answer
86 views

How to prove the impossibility of an equality with floor functions

Given that $r<n$, is it possible to prove that the following equality is impossible for $n>7$ ? (both inequalities are strict here). Some other minimal $n$ would be okay as well. I know it holds ...
13
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2answers
526 views

Prove that $\frac{(3 a+3 b) !(2 a) !(3 b) !(2 b) !}{(2 a+3 b) !(a+2 b) !(a+b) ! a !(b !)^{2}}$ is an integer.

Prove that $$\frac{(3 a+3 b) !(2 a) !(3 b) !(2 b) !}{(2 a+3 b) !(a+2 b) !(a+b) ! a !(b !)^{2}}$$ is an integer for all pairs of positive integers $a, b$ (American Mathematical Monthly) My work - $ v_{...
0
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2answers
58 views

Are there smooth functions that in their limit can perform ceiling and floor operations?

Preamble There is a motivation section at the bottom which explains where this arose from -- might be helpful or of interest There is a set of ideal criteria which if met would define the ideal ...
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2answers
63 views

Is it possible to make velocity time graph periodic?

sorry i know this question is from physics but i believe it uses more of maths. i am a highschool student but here we are not taught Fourier analysis so we can't learn those beautiful curves and ...
2
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1answer
45 views

Floor function parity problem

Prove that for every natural k this expression is always odd $⌊(5+\sqrt{19})^k⌋=A^k$ Progress that I' ve done is: I noticed $9^k<A^k<(9.5)^k$ Also I tried an induction approach, I used Binomial ...
2
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4answers
57 views

For a fixed $k$ what is the value of $\sum_{l=1}^{5^m-1} \Big\lfloor \dfrac{l}{5^k}\Big \rfloor$

For a fixed $k$ what is the value of $\sum_{l=1}^{5^m-1} \Big\lfloor \dfrac{l}{5^k}\Big \rfloor$ By dividing the numbers between $1$ and $5^m$ as intervals of $5^k$, I was getting the following ...
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2answers
32 views

an identity involving the floor function.

Assume that $x$ is a real number and that $m$ and $n$ are positive integers. $[x]$ denotes the greatest integer that is $\leq x$. Can the following identity be true? $$[mnx] = m([nx]+1).$$ I tried to ...
3
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1answer
60 views

Pairwise disjoint sets of the form $\{\lfloor n\alpha \rfloor : n \in \mathbb{N}\}.$

Let $S_{\alpha} = \{\lfloor n\alpha \rfloor : n \in \mathbb{N}\}.$ I am working on a problem which asks to show that $\mathbb{N}$ cannot be partitioned as the pairwise disjoint union of $S_{\alpha}, ...
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0answers
37 views

Why does this relation $\sum ^{k}_{n=m} f( n) =\int\limits ^{k+1}_{m} f(\lfloor x\rfloor ) dx$ not apply here?

I have this relation $\sum ^{k}_{n=m} f( n) =\int\limits ^{k+1}_{m} f(\lfloor x\rfloor ) dx$ I have used this relation before and it was correct in every situation I tried except for this $\int\limits ...
0
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1answer
49 views

What does $[5.9]$ mean?

I came across this notation in the CAA module 0 sample questions. See photo: It looks like it means lower bound but not sure. Can’t find any info. on google either. Anyone come across this notation ...
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0answers
30 views

Ceil and Floor functions on fractions

Is floor(a/k)=ceil(a/(k+1)) always where a>=1 and k>=1? I have tried using the fact that ...
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0answers
50 views

How can we extend this Markov semigroup inequality?

Let $(\kappa_t)_{t\ge0}$ be a Markov semigroup on a measurable space $(E,\mathcal E)$, $$\iota:[0,\infty)\to[0,1)\;,\;\;\;x\mapsto x-\lfloor x\rfloor,$$ $\xi:[0,1]\to[0,1)$ be nonincreasing and $$\...
4
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1answer
51 views

Find all nonconstant polynomials P such that P({X})={P(X)}

Find all nonconstant polynomials $P$ which satisfy $P(\{X\})=\{P(X)\}$, where $\{x\}$ is the fractional part of $x$. I've tried to prove that the polynomial in question is linear, but I can't think of ...
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2answers
58 views

Evaluate $ \lim _{x \rightarrow 0}\left(x^{2}\left(1+2+3+\dots+\left[\frac{1}{|x|}\right]\right)\right) $

Evaluate $$ \lim _{x \rightarrow 0}\left(x^{2}\left(1+2+3+\dots+\left[\frac{1}{|x|}\right]\right)\right) $$ For any real number $a,|a|$ is the largest integer not greater than $a$. I am getting no ...
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1answer
64 views

Compute $\sum_{i=0}^{100}\lfloor i^{3/2}\rfloor+\sum_{j=0}^{1000}\lfloor i^{2/3}\rfloor$

I'm trying to find the value of $\sum_{i=0}^{100}\lfloor i^{3/2}\rfloor+\sum_{j=0}^{1000}\lfloor i^{2/3}\rfloor$. The second sum is obvious, since it's over $j$ instead of $i$. I thought there would ...
1
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3answers
68 views

Last two digits of $[(\sqrt{5}+2)^{2016}]$

I was trying to find the last two digits of the largest integer $\left\lfloor\left(\,\sqrt{\, 5\, }\, +\, 2\,\right)^{2016}\right\rfloor$ less than or equal to $\left(\,\sqrt{\, 5\, }\, +\, 2\,\right)^...
0
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1answer
51 views

If $\xi:[0,1]\to[0,1)$ is decreasing, is $[0,\infty)\ni t\mapsto\xi(1)^{\lfloor t\rfloor}\xi(x-\lfloor x\rfloor)$ eventually nonincreasing?

Let $$\iota:[0,\infty)\to[0,1)\;,\;\;\;x\mapsto x-\lfloor x\rfloor,$$ $\xi:[0,1]\to[0,1)$ be decreasing and $$\tilde\xi(t):=\xi(1)^{\lfloor t\rfloor}\left(\xi\circ\iota\right)(t)\;\;\;\text{for }t\ge0....
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2answers
41 views

Evaluate $\int_{-1}^{3} [ x+ \frac{1}{2}] dx$

Evaluate $\int_{-1}^{3} [ x+ \frac{1}{2}] dx$ where $[.]$ denotes the greatest integer less than or equal to $x$. My attempt : $\int_{-1}^{3} [ x+ \frac{1}{2}] dx= \int_{-1}^{0} [x +1/2]dx + \int_{0}^...
0
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1answer
48 views

Integral of a floor funtion.

Well, i was trying to solve this problem, this told me find the integral \begin{equation} \int_{0}^{2}f(x)dx \end{equation} with $$ x \in <0, \infty>$$ of this funtion: \begin{equation} f(x) =\...
0
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2answers
41 views

Looking for a formula to compute $\left\lceil \frac{x+1}{2} \right\rceil$

I'm looking for a formula to easily compute: $$ \left\lceil \frac{x+1}{2} \right\rceil $$ The formula shouldn't use any floor, ceil or round function. I'm looking for something "simple".
1
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2answers
54 views

Floor function of a product

I'm reading a book about proofs and I'm currently stuck in this problem. Prove that for all real numbers $x$ and $y$ we have that: $$\lfloor x\rfloor \lfloor y\rfloor \leq \lfloor xy\rfloor \leq \...
0
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1answer
34 views

Prove that $\lceil \log_{10}(n) \rceil$ equals the number of digits of $n \ne 10^a$

Prove that $\lceil \log(n) \rceil $ equals the number of digits of $n \ne 10^a$ for some non-negative integer $a$ Let for some positive integer $k$ and non-negative integers $a_i<10$, $i \in \{0,...

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