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).
2,415 questions
5 votes
1 answer
245 views
What's the name for things like $\sin(x)$ and $\lvert x\rvert$ that don't cancel out? Eg, $\sin(x)=\sin(y)$ doesn't imply $x=y$.
What's the name of something where you can't cancel it out? To explain what I mean, $\sin(450^\circ)=\sin(90^\circ)$ doesn't mean that $450=90$ $\lvert17\rvert=\lvert-17\rvert$ doesn't mean $17=-17$ $...
- 51
2 votes
0 answers
63 views
Question regarding a combinatorial map between two sets for proving hermite identity
This is a proof for the hermite identity: $$\sum_{k=0}^{n-1} \left\lfloor x + \frac{k}{n} \right\rfloor=\lfloor nx \rfloor$$ Combinatorial Interpretation of the RHS Let $S$ be the set of positive ...
- 21
0 votes
2 answers
366 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^...
- 405
2 votes
1 answer
158 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
135 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 ...
6 votes
3 answers
270 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}{...
- 3,235
0 votes
2 answers
89 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 ...
- 405
-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
55 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 $$\left\lfloor \dfrac{(n^m+1)^{k+1}}{n^{2m}-1}\right\rfloor-(n^m+1)\left\lfloor \dfrac{(n^m+1)^k}{n^{2m}-1}\right\rfloor = ...
- 127
1 vote
1 answer
99 views
Gnuplot gave me some trouble plotting the Fourier series of $f(x) = e^{-|x|}$. Anyone else have this experience when plotting a Fourier series? [closed]
I wanted a plot of: \begin{equation} f(x) = e^{-|x|} \end{equation} and I wanted to compare $f(x)$ to its Fourier series ($n = 1,3,20$): \begin{equation} F(x) = \frac{e^{\pi}-1}{\pi e^{\pi}} + \frac{2}...
- 39
1 vote
1 answer
72 views
Find the whole part of $E(n)=(\sqrt[3]{n+1}-\sqrt[3]{n}+\sqrt[3]{n-1})^3, n \in \mathbb{Z}$
The Problem Find the whole part of $E(n)=(\sqrt[3]{n+1}-\sqrt[3]{n}+\sqrt[3]{n-1})^3, n\in \mathbb{Z}$ Do the same thing for $\sqrt{}$ instead of $\sqrt[3]{}$ My Idea Use the identity $$(x+y+z)^3=x^3+...
- 405
4 votes
1 answer
72 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 $k \in \mathbb{N}$ , $$U_{n+1} = U_{\color{magenta}n} + \color{red}5$$ $$U_{n+2} = U_{n+1} + \color{red}5$$ ...
- 638
2 votes
0 answers
63 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
277 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=...
- 405
5 votes
3 answers
278 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^...
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