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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2
votes
5answers
117 views

Solving $4\lfloor x \rfloor=x+\{x\}$. What is wrong in my solution? Can anyone tell?

$\lfloor x \rfloor$ = Floor Function, and $\{x\}$ denotes fractional part function Solve for $x$ $$4\lfloor x \rfloor= x + \{x\}$$ $ x = \lfloor x \rfloor + \{x\}$ $\implies x - \{x\} = \lfloor x \...
1
vote
1answer
52 views

Is $\lfloor{\frac{a+b+c+d}{4}}\rfloor=\lfloor\frac{\lfloor{\frac{a+b}{2}}\rfloor+\lfloor{\frac{c+d}{2}}\rfloor}{2}\rfloor$ for $a,b,c,d\in\mathbb R$?

Does the following hold $\forall a,b,c,d\in\mathbb R$? $$\quad\left\lfloor{\frac{a+b+c+d}{4}}\right\rfloor=\left\lfloor\frac{\left\lfloor{\frac{a+b}{2}}\right\rfloor+\left\lfloor{\frac{c+d}{2}}\right\...
1
vote
2answers
20 views

Calculate the minimum value of an integer $x$, such that $\left\lfloor\frac{xy^2}{xy+w(y-z)}\right\rfloor>z$

Given integers: $y>z>0$ $w>0$ I want to calculate the minimum value of an integer $x$, such that $\left\lfloor\frac{xy^2}{xy+w(y-z)}\right\rfloor>z$. I've figured that I can instead ...
0
votes
0answers
43 views

Using trigonometry for the floor and ceiling functions

I was trying to get rid of a floor function from an equation, so I started to play a bit with trigonometry. I found these two replacements, and I would like to know if they are correct. Floor: $$⌊x⌋ = ...
0
votes
0answers
31 views

Ceil function inequality.

I have a function $x:\mathbb{R}_+\to \mathbb{N}$, Im sure that for small enough inputs I can bound this function but I dont know how to make a rigorous argument. Let $a\in\mathbb{R}_+$ and define $$x(\...
3
votes
4answers
77 views

How to solve $\lfloor x \rfloor + \lfloor \frac{1}{x} \rfloor = 1$?

I am stuck with this equation. All I could do is this: $\lfloor x \rfloor$ = $\lfloor n + m \rfloor$ such that $n \in N$ and $m<1$. We get: $\lfloor x \rfloor + \lfloor \frac{1}{x} \rfloor = 1$ $\...
0
votes
1answer
50 views

$ \left\lfloor 10^{\lfloor n \rfloor} \pi \bmod 10 \right\rfloor $ - does this function give the nth decimal place of pi?

Function to round the nth decimal place of pi to the nearest integer. For example, for pi, n = 0, y = 3. n = 1, y = 1. n = 2, y = 4. And so on and so forth. Gives me good results until n = 17, which ...
0
votes
1answer
65 views

How to isolate a variable inside floor and ceiling functions?

I am trying to isolate for the variable $E_c$ in the following equation: $$\Large F_c=Y + \left\lceil\frac{D_3 - Z*Y}{2*Z}\right\rceil$$ where $\large Y=\left\lceil\frac{D_1}{Z}\right\rceil$, $\large ...
0
votes
3answers
61 views

Solving a floor function problem [closed]

I am trying to solve the following question: find all real numbers $x$ such that: $$\Big \lfloor\frac{x^2+1}{10}\Big \rfloor+\Big \lfloor\frac{10}{x^2+1}\Big \rfloor=1$$ If I substitute $n=\frac{x^2+1}...
2
votes
2answers
88 views

Summation on floor function [closed]

If $f(n)=\left[\sqrt{n}+\dfrac{1}{2}\right]$ when $n$ is a natural number and $[\cdot]$ represents the floor function, then find the value of $$\sum_{n=1}^{\infty} \dfrac{2^{f(n)}+2^{-f(n)}}{2^n}$$. ...
3
votes
1answer
53 views

How to isolate a variable within multiple ceiling and floor functions?

I am trying to isolate a variable that is inside a floor function, and this floor function is inside a ceiling function. Here is the equation: $$\Large{F = \left\lceil{256 * D_1\over\left\lfloor\left\...
0
votes
1answer
47 views

How can you find the continuous digits of $g(x)=3 \lfloor x \rfloor^3$?

I have a task that I'm just not getting anywhere with: Let $ g: \mathbb{R} \rightarrow \mathbb{R} $ given by $g(x)=3 \lfloor x \rfloor^3$ for all $ x \in \mathbb{R} $. Determine all places where $g$ ...
2
votes
0answers
129 views

How can I prove $\{a+b\}\{a-c\}\{a-d\}=0$ if the folowing relation holds for all positive integers?

Prove that if are $a, b, c, d$ real numbers for which holds $\lfloor na \rfloor+\lfloor nb \rfloor=\lfloor nc \rfloor+\lfloor nd \rfloor$ for all positive integers $n$ then $\{a+b\}\{a-c\}\{a-d\}=0$. ...
0
votes
2answers
107 views

Inequality involving ceiling of square

If $r$ is such that $r^2$ is an integer, is this expression true for all real $R$? ($r$ and $R$ are always positive) $$ r < R \iff r^2 < \lceil R^2 \rceil $$ It is for programming a script in ...
0
votes
1answer
37 views

Proving nearest integer function identity [closed]

While trying to express the nearest integer function in terms of the modulo or floor function to make some lunar ephemeris equations more compact, I noticed (using GeoGebra) that $$\lfloor x\rceil=1+\...
2
votes
1answer
49 views

Derivative of floor function using epsilon/delta

For every $x\in\mathbb{R}$, let $[x]$ denote the floor of $x$. I read that the derivative of $[x]$ over non-integers is zero, and now I want to show it using epsilon/delta, i.e. show $$0=\lim_{x\...
1
vote
0answers
12 views

Bounding exponentiation functions with floor and ceiling functions as exponents.

Suppose $x \in \mathbb{R}$ and $y,z \in \mathbb{N_{>0}}$, how can one upper bound an exponentiation function $f : \mathbb{R} \times \mathbb{N_{>0}} \times \mathbb{N_{>0}} \to \mathbb{R}$ ...
0
votes
1answer
29 views

finding big omega of $n^k$ by splitting the sum (big-$O$ notation)

I'm currently trying to undestand one of the examples of my discrete mathematics book: Show that: $1^k + 2^k + ··· + n^k = \Theta (n^{k+1})$ I have no trouble finding big - O, but rather finding the ...
0
votes
1answer
33 views

Prove integer remains $\mathbb{Z}_{n}$ after Division and Floor

Given a positive integer $w$, let $r$ be an integer that satisfies: $r \in{} \{0,1,2, \ldots{}, 2^{w}-1 \}$, that is, $r \in{} \mathbb{Z}_{2^w}$. I want to prove that: for any $p$, that $p \in{} \...
0
votes
1answer
61 views

How can one find the x value that gives the largest possible value to the equation (100floor(10000/(floor(100000/(2x))+1)))/x?

How can one find the x value that gives the largest possible value to the equation $$\frac{100\left\lfloor\frac{10000}{\left\lfloor \frac{100000}{2x}\right\rfloor+1}\right\rfloor}{x}$$ where $x$ is a ...
0
votes
1answer
29 views

Continuity of composition of root and floor function

Can someone give me a hint to prove the continuity of the following function: $f: \mathbf{R}\to\mathbf{R}, f(x):=\sqrt{\lfloor{x^2}\rfloor}$. I already proved the continuity of the root function and ...
1
vote
5answers
102 views

Number of real solutions of $\begin{array}{r} {\left[\frac{2 x+1}{3}\right]+\left[\frac{4 x+5}{6}\right]} =\frac{3 x-1}{2} \end{array}$

Solve for $x \in \mathbb{R}$ $$\begin{array}{r} {\left[\frac{2 x+1}{3}\right]+\left[\frac{4 x+5}{6}\right]} =\frac{3 x-1}{2} \end{array}$$ where $[x]$ denotes greatest integer less than or equal to $...
0
votes
1answer
40 views

Solving a exponential floor flunction equation

I am trying to solve this question but can't simplify this further. Question: $$ 2^{\lfloor log_2{(x)} + \frac{1}{2} \rfloor} = 2^{\lfloor log_2(x-2^{\lfloor{ log_2{(\frac{x}{2})} + \frac{1}{2}}\...
2
votes
3answers
97 views

How do we solve $x^2 + \{x\}^2 = 33$ without computer?

This is a problem taken from a group on Facebook. I wonder how to solve this without numerical process. $x^2 + \{x\}^2 = 33\tag{1}$ My unfinished attempt: $$\begin{align} x^2 + \{x\}^2 &= 33\\ x^2 ...
0
votes
0answers
36 views

Floor function for power of 2

I am confused about this. Which one is true? $2^{\lfloor{2log_2(x)}\rfloor} =\lfloor{x^2}\rfloor$ or $2^{\lfloor{2log_2(x)}\rfloor} =\lfloor{x}\rfloor^2$
1
vote
1answer
54 views

Trigonometric terms for floor function $Q_k(n)$

I'm working on some problems of number theory and somehow I could manage to find a more general formula for some problems. However, I needed to define a function $$Q_k(n) = \text{floor}(\frac n{k}) \...
1
vote
1answer
53 views

Rigorous definition of the greatest integer function

I am sorry, if the title wasn’t very clear and concise, but I would try to explain my question here more elaborately. I am learning elementary number theory, which is when I came across this function. ...
0
votes
3answers
63 views

Proof that $\left\lfloor -x\right\rfloor =-\left\lceil x\right\rceil$

I wanted to ask if this kind of reasoning for proving the result in the title could be considered correct: We know that: $\left\lceil x\right\rceil =n$ if and only if $n-1<x\leq n$ Then $-\left\...
6
votes
2answers
231 views

Evaluate the integral : $\int_0^\infty (-1)^{ \lfloor x \sin x \rfloor } dx$

How to evaluate the following integral? $$ \int_0^\infty (-1)^{ \lfloor x \sin x \rfloor } dx$$ I have no idea how to calculate this improper integral. Maybe I have to use some property of floor ...
9
votes
3answers
157 views

$\{x^2\} = \{x\}^2$, how many solutions in interval $[1, 10]$

Find how many solutions there are in the interval $[1, 10]$ to the fractional part equation: $$\left\{x\right\}^2 = \left\{x^2\right\}$$ Where $\{\cdot\}$ is the fractional part function, meaning that:...
8
votes
2answers
76 views

Find all positive integers $n$, such that $(\left\lfloor \sqrt{n} \right\rfloor^{2} +2) | (n^2 + 1) $

I tried to look at the cases when $n$ is a perfect square. Then $\left\lfloor \sqrt{n} \right\rfloor^{2} +2= n+2$, $ n^2 + 1 =(n-2)(n+2) + 5$. Then we must have $(n+2)|5$. But only $1$ and $5$ divide $...
2
votes
2answers
54 views

Is this floor division expansion true for all positive integers?

Suppose you have 3 positive (non-zero) integers $n,x,y$. Is the following statement true? $$ \left\lfloor\frac{n y}{x}\right\rfloor = y\left\lfloor\frac{n}{x}\right\rfloor+\left\lfloor\frac{y\left(n\...
1
vote
1answer
40 views

Find the set of natural numbers $n \in \mathbb{N}$ for which $\lfloor\sqrt{n+1}\rfloor \neq\lfloor\sqrt{n}\rfloor$

I have to find the set of natural numbers $n \in \mathbb{N}$ for which $\lfloor\sqrt{n+1}\rfloor \neq\lfloor\sqrt{n}\rfloor$. I have tried writing the formal definition of the floor function and tried ...
2
votes
1answer
53 views

Isn't $\lceil \frac{xy}{z}\rceil =\lfloor \frac{xy+y-1}{z}\rfloor$ for any positive integers $x,y,z$?

I was pretty sure that: $$\left\lceil\frac{xy}{z}\right\rceil = \left\lfloor\frac{xy+y-1}{z}\right\rfloor$$ for positive integers $x,y,z$. But I'm getting wrong results testing it in Python 3: ...
1
vote
1answer
46 views

Why this approach works for solving USAMTS's number theory problem?

This is the first problem of USAMTS $1998$ Round $1$ Several pairs of positive integers $(m ,n )$ satisfy the condition $19m + 90 + 8n = 1998$. Of these, $(100, 1 )$ is the pair with the smallest ...
-1
votes
3answers
77 views

Prove $\lfloor x\rfloor + \lfloor y\rfloor = \lfloor x + \lfloor y\rfloor\rfloor$ [closed]

I am unable to prove a very interesting floor identity: $$\lfloor x\rfloor + \lfloor y\rfloor = \lfloor x + \lfloor y\rfloor\rfloor$$
0
votes
0answers
59 views

Proving an equality with floor functions

I have a certain equation derived from another problem I was solving that includes floor functions. By plugging in different values I experimentally find that the following equation is true: $\left \...
1
vote
2answers
87 views

Find the minimum natural number $n$, such that the equation $\lfloor \frac{10^n}{x}\rfloor=1989$ has integer solution $x$

Find the minimum natural number $n$, such that the equation $\lfloor \frac{10^n}{x}\rfloor=1989$ has integer solution $x$. My work- $\frac{10^n}{x}-1<\lfloor \frac{10^n}{x}\rfloor≤\frac{10^n}{x}\...
1
vote
0answers
89 views

How does this $\lfloor{\frac{qm[i] + \lfloor\frac{t+1}{2}\rfloor}{t}\rfloor}$ round up in case of a tie?

There is a code that is supposed to multiply a vector by $\lfloor q/t\rfloor$. It says this: ...
0
votes
0answers
12 views

partly piecewise increasing Floor function, ask for value in certain condition

A Maths Olympiad problem we have $f(n) = \lfloor\frac{n}{\lfloor\sqrt{n}\rfloor}\rfloor$ try find for all n (and prove) such that $f(n)< f(n+ 1)$, $f(n) > f(n+ 1)$, $f(n)= f(n+ 1)$. with n ...
13
votes
3answers
843 views

Find the two last digits of $[(29+\sqrt{21})^{2000}]$.

Let $n=\left[(29+\sqrt{21})^{2000}\right]$. I want to find the two last digits of $n$. I known that, $S_k=(29+\sqrt{21})^{2000}+(29-\sqrt{21})^{2000}$ is integer, but from this I could not find the ...
1
vote
1answer
44 views

Integer part problem

If $m,n$ are natural non-zero numbers show that $$[x]+[x+1/n]+[x+2/n]+...+[x+m/n]=[nx]$$ for any real $x$ if and only if $m=n-1$. $[x]$ is the integer part of $x$. I know from the Hermite Identity ...
1
vote
4answers
107 views

Struggling to solve an equation containing a floor function

I am trying to solve the following equation ($x\in \mathbb R)$ $$ 2\lfloor{(x+1)^2+8}\rfloor=(x+1)(2x+3) \tag{1} $$ I have tried substituting $x$ with its split form (integer part ($n$) + decimal part ...
-1
votes
1answer
38 views

Is this equation about floor function correct?

I am trying to solve this equation. Please correct me. $$\lfloor{x}\rfloor + \lfloor{w} - \lfloor{x}\rfloor \rfloor=\lfloor{x}\rfloor-\lfloor{x}\rfloor+\lfloor{w}\rfloor$$ $$=\lfloor{w}\rfloor$$
0
votes
0answers
21 views

How to formulate the whole number part of decimal number in equation

I would like to formulate the classification with equal interval method. I know how to formulate the interval: $$ \frac{(Highest\;Value - Lowest\;Value)}{Number\;of\;Classes}=Interval\;Value $$ But ...
2
votes
0answers
86 views

Number theory Question From RMO(2015) regarding fractional part function. [duplicate]

Show that there are infinitely many positive real numbers $a$ which are not integers such that $a(a-3\{a\})$ is an integer (here $\{a\}$ denotes the fractional part of $a$). I tried putting $a = [a]+\{...
1
vote
2answers
78 views

Prove that $\lim_{x \to 0+}\lfloor{\frac{x}{a}\rfloor}\frac{b}{x}=0$

So, I gotta prove that $\lim_{x \to 0+}\lfloor{\frac{x}{a}\rfloor}\frac{b}{x}=0$, but my only info is that $a,b \in \mathbb{R_{+}^{*}}$. I assumed $\lfloor{\frac{x}{a}\rfloor}=\frac{x}{a}-[\frac{x}{a}]...
0
votes
1answer
36 views

Better way to compute this definite piecewise integral

I have to find the value of $$I = \int_0 ^{10} [x]^3\{x\}dx$$ Where $[x]= $greatest integer less than or equal to$ x $(the greatest integer function or the floor function) And $\{x\}= $fractional part ...
-2
votes
1answer
64 views

$\forall\,x,\,y\in\mathbb{R} : [x + y] = [x] + [y]$ [closed]

Can you please help me proving the identity $\forall\,x,\,y\in\mathbb{R} : [x + y] = [x] + [y]$, where $[\alpha]$ means the integer part of $\alpha$? I figured out that it holds for $x,\,y\in\mathbb{...
2
votes
2answers
99 views

Proving $\left\lfloor \sqrt{x^2+3x+3}\right\rfloor = x+1$

Let $$f(x)=\sqrt{x^2+3x+3}$$ Claim: $$\lfloor f(x)\rfloor = x+1$$ I came across this expression in a question and made this claim based on observations $$⌊f(1)⌋=\lfloor\sqrt{7}\rfloor=2$$ $$⌊f(2)⌋=\...

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