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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7
votes
1answer
80 views

$\sum_{i=0}^k \lfloor\sqrt{ip} \rfloor = \frac{(p^2-1)}{12}$ where p is a prime and $p=4k+1$

Question: Let p be a prime number of the form 4k+1. Show that $\sum_{i=0}^k \lfloor\sqrt{ip} \rfloor = \frac{(p^2-1)}{12}$ Source: I came across this question while solving An Introduction to the ...
1
vote
0answers
41 views

Simplifying expression with ceil

How can I simplify the following? $\lceil \frac{x}{\lceil \frac{x}{y} \rceil}\rceil$ I'm dealing with partitioning data for processing, where x is the total input size and y is the maximum chunk size ...
3
votes
2answers
44 views

Find solutions for integrals with floor function

Find solutions of x satisfying: $$\int_{0}^{2\lfloor{x+14}\rfloor}\left(\frac{t}{2}-\left\lfloor{\frac{t}{2}}\right\rfloor\right)dt=\int_{0}^{x-\lfloor{x}\rfloor}\lfloor{t+14}\rfloor dt$$ My solution ...
0
votes
0answers
38 views

Find $\{u \geq a\}$ for all $a \in \mathbb{R}$, $u(x)=\lfloor x\rfloor$

Let $u: \mathbb{R} \rightarrow \mathbb{R}$, $u(x)=\lfloor x\rfloor$. (i) Determine $\{u \geq a\}$ for all $a \in \mathbb{R}$. (ii) Show that $u$ is $\mathcal{B}(\mathbb{R}) / \mathcal{B}(\mathbb{R})$-...
0
votes
2answers
35 views

How to integrate : $\int_{-1}^{\frac{3}{2}} \left[x^2\right] dx$ where [ ] represents Greatest Integer Function

Please explain to me this question. How to integrate: $$\int_{-1}^{\frac{3}{2}} \left[x^2\right] dx$$ where $[ \ \ ]$ represents Greatest Integer Function? I am not able to understand how do come to ...
0
votes
1answer
18 views

Checking if there is a better way of handling the distribution of $K\times\lfloor \frac{X}{K}\rfloor$ given $X$ is already known.

Suppose we have a random variable $X$ and we know its distribution. How would we determine the distribution of $Y:=K\times\lfloor \frac{X}{K}\rfloor$ in relation to $X$, where $K$ is just any constant?...
3
votes
1answer
32 views

residues mod. n of $\lfloor k\alpha\rfloor$ where $\alpha$ is irrational positive.

Consider for any integer $k\geqslant1$ : $$u_k=\lfloor k\sqrt2\rfloor$$ It's not difficult to prove that there are infinitely many $k$ such that $u_k$ is even and infinitely many $k$ such that $u_k$ ...
1
vote
2answers
62 views

Given $x^5-x^3+x-2=0$, find $\lfloor x^6\rfloor$. [duplicate]

If $\alpha$ is a real root of the equation $x^5-x^3+x-2=0$, find the value of $\lfloor\alpha^6\rfloor$, where $\lfloor x\rfloor$ is the least positive integer not exceeding $x$. My approach is to ...
3
votes
3answers
56 views

How many positive integer solutions exist for $[\frac{x}{19}]=[\frac{x}{20}]$, where $[x]$ denotes the Greatest integer function

Question How many positive integer solutions exist for $[\frac{x}{19}]=[\frac{x}{20}]$, where $[x]$ denotes the Greatest integer function What I tried I took the following cases one by one, CASE $1$ $$...
1
vote
4answers
64 views

What's the value of $\lim_{n\to \infty} \frac{n+1}{n^2+1}\left[\frac{n^2+1}{n+1}\right]$

Excuse my bad typing of mathematical symbols, I don't know how to type symbols. The $[ ]$ brackets are for the floor function where $[3.2]=3$ and $[2.9]=2$. I have trouble calculating this limit, I ...
2
votes
3answers
89 views

Floor function bounding

From CMC: What is the sum of the square of the real numbers $x$ for which $x^2 - 20\lfloor x\rfloor + 19 = 0$? We use $\lfloor x\rfloor\le x<\lfloor x\rfloor+1$ and eventually get the bounds $1\...
1
vote
1answer
27 views

Show that $\lceil n\cdot log_ab\rceil$ and $\lceil n\cdot log_\frac{b}{a}b\rceil$ “covers” all integers

Can it be shown that $\lceil n\cdot log_ab\rceil$ and $\lceil n\cdot log_\frac{b}{a}b\rceil$ are "complementary" ($1<a<b$ , $b$ is not a power of $a$) ? By "complementary" I ...
1
vote
0answers
37 views

Sequences and Greatest Integer Function

$u_1 = 1$ $u_{n+1} = u_n + 1/u_n$ for $n > 1$. Find $\lfloor 50u_{100}\rfloor$, where $\lfloor x\rfloor$ represents the greatest integer less than or equal to $x$. Kindly give any clues/ideas on ...
5
votes
1answer
81 views

Find the probability that $[x+y+z]=[x]+[y]+[z]+2$

Find the probability that the equation $[x+y+z]=[x]+[y]+[z]+2$ is true, where $x,y,z \in R$. [.] Represents the greatest integer function. I got two different answers by two different methods. 1st ...
1
vote
0answers
42 views

Sequence defined by recursive formula and floor function

Can someone help with this question: Let $a_0,a_1,a_2,...$ be a sequence of real numbers satisfying $a_0=1$ and $a_n=a_\left\lfloor{\frac{7n}{9}}\right \rfloor + a_\left\lfloor{\frac{n}{9}}\right\...
2
votes
2answers
103 views

Is there a simpler expression for this piecewise-defined function?

As a math-for-fun exercise, I challenged myself to find a globally-defined, everywhere-differentiable antiderivative of $\sqrt{1-\sin(x)}$; this boils down to evaluating the integral $f(x)=\int_{-\...
0
votes
0answers
19 views

Graph of $(\lfloor(f(x))\rfloor)^2$

How do I plot $(\lfloor x^2 \rfloor)^2$? I know how to plot $\lfloor x^2 \rfloor$ from $x^2$. However, where do I begin with in plotting $(\lfloor x^2 \rfloor)^2$ from$\lfloor x^2 \rfloor$? Or more ...
5
votes
2answers
106 views

$[n \sqrt{2}] = [m (2+\sqrt{2})]$ for $m,n$ natural.

Does $[n \sqrt{2}] = [m (2+\sqrt{2})]$ for $m,n$ natural have no solution where $[x]$ is the floor function of $x$? I tried calculating some examples ($1,000,000$ examples on Python) and it seems as ...
3
votes
4answers
95 views

How to solve $ \sum_{i=1}^{n} \left \lfloor{\log{i}}\right \rfloor $ for closed form

I'm trying to get a closed form of this equation: $$ \sum_{i=1}^{n} \left \lfloor{\log{i}}\right \rfloor $$ I know that $$ \sum_{i=1}^{n} {\log{i}} = \log{n!}$$ But I'm confused about how the floor ...
2
votes
1answer
23 views

Divide an integer in the sum of two integers with percentage factor using ceil and floor

I have encoutered a problem in a software that I use for invoicing. I have a variable (quantity) integer A which I want to split in a sum of two integers using a percentage p where $A1 = p*A$ and $A2 =...
1
vote
3answers
72 views

Find a limit involving floor function

I had to find the following limit: $\displaystyle{\lim_{x \to \infty}}\frac{x}{\lfloor x \rfloor}$ Where $x\in\mathbb{R}$ and $f(x)= {\lfloor x \rfloor}$ denotes the floor function. This is what I did:...
3
votes
2answers
104 views

Simpler ways of finding solutions to $\int_0^x \lfloor{x\rfloor}^2 dx=2(x-1)$

So I was solving the following question: Find the number of solutions to $$\int_0^x \lfloor{x\rfloor}^2 dx=2(x-1)$$ for $x<0$.$$$$ Options: $2,3,4,5$ And I managed to show that there's no ...
0
votes
3answers
71 views

How to solve these inequalities with floors and ceilings?

I am trying to solve these two inequalities for $x$, as a function of the positive integer $n$: $$ \newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor} \newcommand{\ceil}[1]{\left\lceil #1 \right\...
-1
votes
2answers
50 views

Simplify: $x\leq\lfloor\frac{2(y+2x)+1}{5}\rfloor $

I want to simplify this term: $$x\leq\left\lfloor\frac{2(y+2x)+1}{5}\right\rfloor $$ Any help would be most appreciated.
0
votes
1answer
24 views

$Z$-transform of floor($\frac{n}{5}$) [closed]

I found problem about floor array that I can't solve. Find Z transform of f(n) = floor(n/5). I tried writing this array and for n from 1 to infinity, I got n/5 = 1/5 + 2/5 + 3/5 + 4/5 + 5/5 + 6/5... ...
1
vote
1answer
71 views

Advanced Inequality with Floor Functions [duplicate]

For positive real numbers $a,$ $b,$ $c,$ and $d,$ find the minimum value of $$\left\lfloor \frac{b + c + d}{a} \right\rfloor + \left\lfloor \frac{a + c + d}{b} \right\rfloor + \left\lfloor \frac{a + b ...
4
votes
2answers
73 views

finding bounds for $\int_0^X\lfloor x^2\rfloor \, dx$

I am trying to find bounds for: $$I=\int_0^X\lfloor x^2\rfloor \, dx$$ and through integrating two different ways I ended up with the sums: $$I=(X^2+1)^{3/2}-\sum_{i=1}^{X^2+1}\sqrt{i}$$ $$I=X^3-\sum_{...
1
vote
2answers
51 views

Integrating floor functions without known limits

Suppose we have$$\frac{\displaystyle\int_0^n{\lfloor x \rfloor}\,dx}{\displaystyle\int_0^n{\lbrace x \rbrace }\,dx}$$ $n \in I$ where $\lfloor \cdot\rfloor$ and $\lbrace \cdot\rbrace $ represent the ...
-5
votes
0answers
37 views

Function missing squares. [duplicate]

Source: Taken from the math-olympiad textbook of A.Engel : Show that $$\bigg \lfloor{n+ \sqrt{n}+ \frac{1}{2}}\bigg \rfloor $$ is never equal to a square.
0
votes
4answers
49 views

Addition and Multiplication of Step Function?

Let $f(x) = [x]$ and let $g(x) = [2x]$ for all real $x$. In each case, draw the graph of the function h defined over the interval $[-1,2]$ by the formula given. 1.$h(x) = f(x) + g(x)$ 2.$h(x) = f(x)g(...
0
votes
1answer
50 views

Let $f(x) = \lfloor x \lfloor x \rfloor \rfloor$ for $x \ge 0.$ Find all $x \ge 0$ such that $f(x) = 5.$

Let $f(x) = \lfloor x \lfloor x \rfloor \rfloor$ for $x \ge 0.$ Find all $x \ge 0$ such that $f(x) = 5.$ I found that $f(2)=4$ and $f(3)=9$ so $f(x)=5$ should be in $x \in (2,3).$ But I found that $f(...
0
votes
2answers
52 views

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

Let $f(x) = \lfloor x \lfloor x \rfloor \rfloor$ for $x \ge 0.$ Find the number of possible values of $f(x)$ for $0 \le x \le 10.$ I split the problem into three cases, $1 \leq n < 10, n=10,$ and $...
0
votes
1answer
34 views

Let $N = \sum_{k = 1}^{1000}k(\lceil \log_{\sqrt {2}}k\rceil - \lfloor \log_{\sqrt {2}}k \rfloor).$ Find $N$.

Let$$N = \sum_{k = 1}^{1000}k(\lceil \log_{\sqrt {2}}k\rceil - \lfloor \log_{\sqrt {2}}k \rfloor). $$ Find $N$. I'm not sure how to start this problem. Could someone help me out? Thanks!
2
votes
1answer
66 views

If $f(x)=\big\lfloor x\lfloor x\rfloor\big\rfloor$ for all $x\geq 0$, then for an integer $n$, solve for $x\geq 0$ such that $f(x)=n$.

Let $f(x) = \big\lfloor x \lfloor x \rfloor \big\rfloor$ for $x \ge 0.$ (a) Find all $x \ge 0$ such that $f(x) = 1.$ (b) Find all $x \ge 0$ such that $f(x) = 3.$ (c) Find all $x \ge 0$ such that $f(x) ...
1
vote
0answers
161 views

Proof: $\lfloor\sqrt{a_1}\rfloor+\lfloor\sqrt{a_2}\rfloor+ \dots+\lfloor\sqrt{a_{25}}\rfloor\ge\lfloor\sqrt{a_1+a_2+\dots+a_{25}+200k}\rfloor$ [duplicate]

Source: Proof: $[\sqrt {{a_1}} ] + [\sqrt {{a_2}} ] + \dots + [\sqrt {{a_{25}}} ] \ge \left[ {\sqrt {{a_1} + {a_2} + \cdots + {a_{25}} + 200k} } \right].$ Let $a_1,a_2,\dots,a_{25}$ be nonnegative ...
4
votes
2answers
75 views

Showing that $S_n -\lfloor S_n \rfloor \sim U[0,1]$

$k \in \mathbb{N}$ is fixed $(X_n)_{n \geq 1}$ are all independent and follow an uniform law on $[0,k]$ We define $f(x)=x -\lfloor x \rfloor$ $S_n= \sum_{i=1}^{n} X_i$ $Z_n= f(S_n)$ We want to show ...
0
votes
2answers
45 views

Find all $x$ for which $\left| x - \left| x-1 \right| \right| = \lfloor x \rfloor.$ [closed]

Find all $x$ for which $$\left| x - \left| x-1 \right| \right| = \lfloor x \rfloor.$$Express your answer in interval notation. I started by looking at x < 1 and x $\geq$ 1 separately, but I got ...
1
vote
1answer
29 views

rounding numbers to 10ths, 100ths or 1000ths

I was going through the C++ 10th edition deitel's book. There I encountered a questioned regarding floor() function. And here's the question: Floor() practice question Now, from what I understand so ...
1
vote
2answers
43 views

Find the number of points of discontinuity for $[\cot^{-1} x]$ [.] is the floor function

The range of $\cot ^{-1} $ is $[-\frac{\pi}{2},\frac{\pi}{2}]$ So the range for $[\cot ^{-1} ]$ is $\{-2,-1,0,1\}$ So there must be 4 points of discontinuity, but the answer says there are only three. ...
3
votes
2answers
77 views

How to find range of $\left[\frac{[x]}{x}\right]$

If $[.]$ denotes greatest integer function, find the range of $$\left[\frac{[x]}{x}\right]$$ My friend and I tried solving this question and arrived at the answer ${0,1}$ but when we took the value ...
15
votes
1answer
286 views

How to prove that for $a_{n+1}=\frac{a_n}{n} + \frac{n}{a_n}$ , we have $\lfloor a_n^2 \rfloor = n$?

Let $(a_n)_{n\ge 1}$ be the sequence defined as the following : $$a_1=1 ,\ a_{n+1}=\dfrac{a_n}{n} + \dfrac{n}{a_n} ,\ n\ge1$$ Show that for every $n\ge4,\ \lfloor a_n^2 \rfloor = n$. My approach to ...
1
vote
1answer
69 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\...
2
votes
1answer
35 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
votes
2answers
81 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 ...
1
vote
1answer
43 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\...
0
votes
2answers
57 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
votes
1answer
92 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
votes
1answer
71 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\...
4
votes
0answers
62 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 ...
12
votes
3answers
273 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 ...

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