All Questions
Tagged with ceiling-function or ceiling-and-floor-functions
2,413 questions
0
votes
2
answers
360
views
Solve $[x]+[x^2]=[x^3]$
the problem
Solve $[x]+[x^2]=[x^3]$
my idea
using the fact that $ x=[x]+ \{ x \} $ we can write the equation as $x^3-x^2-x=\{x^3\}-\{x^2\}-\{x\} \in (-2,1)$ because ${x} \in [0,1)$
Now we can solve $x^...
- 539
2
votes
1
answer
151
views
Limit with floor sums reminiscent of the exponent of the central binomial coefficient
This problem comes from the 1976 Putnam exam.
Evaluate
$$
L=\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n
\left(
\left\lfloor\frac{2n}{k}\right\rfloor
-2\left\lfloor\frac{n}{k}\right\rfloor
\right),
$$
...
- 23
3
votes
2
answers
131
views
Prove that $\sum_{k=1}^{n} \left\lfloor \log_{2}\!\left(\frac{2n}{2k-1}\right) \right\rfloor = n$ where $n$ is a natural number.
How can I prove that $\sum_{k=1}^{n} \left\lfloor \log_{2}\!\left(\frac{2n}{2k-1}\right) \right\rfloor = n$, where $n$ is a natural number?
I discovered this identity while trying to prove Prove using ...
3
votes
1
answer
135
views
How to prove : $\{ (2+\sqrt3)^n\} = 1 -(2-\sqrt3)^n$
I need help proving that for all n positive integer $$\{ (2+\sqrt3)^n\} = 1 -(2-\sqrt3)^n$$
where $\{ x \}$ is the fractional part.
We have
\begin{align*}
(2+\sqrt3)^n &= \sum_{k=0}^{n} \binom{n}{...
- 3,155
0
votes
2
answers
86
views
Determine $x,y \in \Bbb{N}$ with the property that $[\sqrt[n]{x^n+y}]=[\sqrt[n]{y^n+x}]$
the problem
Let $n \in \Bbb{N}, n\geq 2$. Determine $x,y \in \Bbb{N}$ with the property that $[\sqrt[n]{x^n+y}]=[\sqrt[n]{y^n+x}]$, where $[a]$ represents the integer part of the real number $a$.
my ...
- 539
-1
votes
1
answer
79
views
If $k < n^m-1$ it follows that expression is $k$
If $n,m,k \in \mathbb{N}$ must be such that $k < n^m-1$ it follows that $$\left\lfloor \dfrac{(n^m+1)^{k+1}}{n^{2m}}\right\rfloor-(n^m+1)\left\lfloor \dfrac{(n^m+1)^k}{n^{2m}}\right\rfloor = k.$$
...
- 127
1
vote
1
answer
54
views
If $2^k < n^m-1$ it means that expression is $2^{k-1}$
If $n,m,k \in \mathbb{N}$ must be such that $2^k < n^m-1$ it follows that $$\left\lfloor \dfrac{(n^m+1)^{k+1}}{n^{2m}-1}\right\rfloor-(n^m+1)\left\lfloor \dfrac{(n^m+1)^k}{n^{2m}-1}\right\rfloor = ...
- 127
1
vote
1
answer
97
views
Gnuplot gave me some trouble plotting the Fourier series of $f(x) = e^{-|x|}$. Anyone else have this experience when plotting a Fourier series? [closed]
I wanted a plot of:
\begin{equation}
f(x) = e^{-|x|}
\end{equation}
and I wanted to compare $f(x)$ to its Fourier series ($n = 1,3,20$):
\begin{equation}
F(x) = \frac{e^{\pi}-1}{\pi e^{\pi}} + \frac{2}...
- 39
1
vote
1
answer
71
views
Find the whole part of $E(n)=(\sqrt[3]{n+1}-\sqrt[3]{n}+\sqrt[3]{n-1})^3, n \in \mathbb{Z}$
The Problem
Find the whole part of $E(n)=(\sqrt[3]{n+1}-\sqrt[3]{n}+\sqrt[3]{n-1})^3, n\in \mathbb{Z}$ Do the same thing for $\sqrt{}$ instead of $\sqrt[3]{}$
My Idea
Use the identity
$$(x+y+z)^3=x^3+...
- 539
4
votes
1
answer
71
views
A similar sequence to OEIS-A098021
Let $U_0=0$, we make $U_n$ by the following rules :
If $\color{magenta}n=U_k$ for some $k \in \mathbb{N}$ ,
$$U_{n+1} = U_{\color{magenta}n} + \color{red}5$$
$$U_{n+2} = U_{n+1} + \color{red}5$$
...
- 604
2
votes
0
answers
60
views
Intervals of continuity of $ f(x)=\frac{\Gamma(x+1)}{\lfloor\Gamma(x+1)\rfloor+1}$
Let
$$\Gamma(x)=\int_0^\infty t^{x-1}e^{-t}\,dt.$$
Define
$$f(x)=\frac{\Gamma(x+1)}{\lfloor\Gamma(x+1)\rfloor+1},\qquad x\in[0,\infty).$$
Find the maximal open intervals $I_n=(a_n,b_n)\subset[0,\infty)...
- 64
1
vote
2
answers
266
views
Determine the smallest possible value of the natural number $ a_1$
Determine the smallest possible value of the natural number $ a_1$, knowing that there exist natural numbers
$ a_1 \geq a_2 \geq \ldots \geq a_{100} \geq 2 $ with the property that
$$
\left\{ \sum_{k=...
- 539
5
votes
3
answers
267
views
(INMO 2009) Find all real numbers $x$ such that $\lfloor x^2 + 2x\rfloor = \lfloor x\rfloor^2 +2\lfloor x\rfloor$.
First of all, the notation $\lfloor x\rfloor$ is the greatest integer not exceeding $x$. That is $\lfloor x\rfloor = \max\{m \in \mathbb{Z}: m\leq x\}$.
Question: Find all real solutions of $\lfloor x^...
user1598427
3
votes
3
answers
148
views
Solving $[x/2]+[x/3]+[x/4]=x$, where [.] denotes GIF
In the equation $$[x/2]+[x/3]+[x/4]=x$$, where [.] denotes GIF/Floor function,
$x$ has to be an integer.
Next, by estimation we have
$$x/2-1<[x/2]\le x/2$$
$$x/3-1<[x/3]\le x/3$$
$$x/4-1<[x/4]...
- 47.2k
0
votes
1
answer
72
views
Does every $r \geq 1$ give a different power sequence? [closed]
Let $r$ be a real number greater than or equal to $1$. I define the positive integer power sequence associated with $r$ to be the sequence of floors of $n^r$, where $n$ is a positive integer. For ...
- 24.8k