The numbers of irreducible fractions is even. For example: if $n=6$,
$\frac{1}{6},\frac{2}{6},\frac{3}{6},\frac{4}{6},\frac{5}{6}$. Here, $\frac{1}{6}$ and $\frac{5}{6}$ is irreducible and there number is tow, in fact even.
Search found 98 matches
- Tue Apr 18, 2017 8:28 pm
- Forum: Junior Level
- Topic: Beginner's Marathon
- Replies: 68
- Views: 44491
- Tue Apr 18, 2017 7:06 pm
- Forum: Combinatorics
- Topic: China Girls MO 2016/6
- Replies: 0
- Views: 4889
China Girls MO 2016/6
Find the greatest positive integer $m$, such that one of the $4$ letters $\text{C, G, M, O}$ can be placed in each cell of a table with $m$ rows and $8$ columns, and has the following property: For any two distinct rows in the table, there exists at most one column, such that the entries of these tw...
- Tue Apr 18, 2017 7:02 pm
- Forum: Junior Level
- Topic: Beginner's Marathon
- Replies: 68
- Views: 44491
Re: Beginner's Marathon
$\text{Problem 15}$
Let n be an integer greater than $2$. Prove that among the fractions
$$\frac{1}{n},\frac{2}{n}\cdots ,\frac{n-1}{n}$$
an even number are irreducible.
Let n be an integer greater than $2$. Prove that among the fractions
$$\frac{1}{n},\frac{2}{n}\cdots ,\frac{n-1}{n}$$
an even number are irreducible.
- Tue Apr 18, 2017 6:56 pm
- Forum: Junior Level
- Topic: Beginner's Marathon
- Replies: 68
- Views: 44491
Re: Beginner's Marathon
$\text{Solution to 14}$ Consider the dilation with center $A$ that carries the incircle to an excircle. The diameter $DT$ of the incircle must be mapped to the diameter of the excircle that is perpendicular to $BC$. It follows that $T$ must get mapped to the point of tangency between the excircle an...
- Tue Apr 18, 2017 12:04 am
- Forum: Social Lounge
- Topic: BDMO Forum Mafia #2
- Replies: 30
- Views: 47750
Re: BDMO Forum Mafia #2
1. ahmedittihad
2. Raiyan Jamil
3. dshasan
4. Epshita32
5. Ananya Promi
6. TashkiManda
7. nahinmunkar
8. Thamim Zahin
2. Raiyan Jamil
3. dshasan
4. Epshita32
5. Ananya Promi
6. TashkiManda
7. nahinmunkar
8. Thamim Zahin
A Lemma?
In a $\triangle ABC$, let $H_A,H_B,H_C$ be the projection of $A,B,C$ on $BC,CA,AB$ respectively. And $O$ is the circumcenter
1. Prove that, $OA \perp H_BH_C$.
2. Let a line $\lambda$ be tangent to $(HBC)$ at $H$. Prove that, $OA \perp \lambda$.
1. Prove that, $OA \perp H_BH_C$.
2. Let a line $\lambda$ be tangent to $(HBC)$ at $H$. Prove that, $OA \perp \lambda$.
- Wed Apr 12, 2017 2:31 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO 2017 junior/4
- Replies: 11
- Views: 7904
Re: BdMO 2017 junior/4
Chinese remainder theoremjayon_2 wrote:what is crt?ahmedittihad wrote:Apply crt to get $n=585$.
http://s3.amazonaws.com/aops-cdn.artofp ... theory.pdf
Take a look at Page no.40 for further information.
- Mon Apr 10, 2017 7:14 pm
- Forum: Junior Level
- Topic: Beginner's Marathon
- Replies: 68
- Views: 44491
Re: Beginner's Marathon
$\text{Problem 12}$
For every positive integer $k$ let $f(k)$ be the largest integer such that $2^{f(k)} \mid k$. For every positive integer $n$ determine : $f(1) + f(2) +f(3)+f(4)+f(5)+\cdots+ f(2^n)$.
Or, $\displaystyle\sum_{i=1}^{2^n} f(i)=?$
For every positive integer $k$ let $f(k)$ be the largest integer such that $2^{f(k)} \mid k$. For every positive integer $n$ determine : $f(1) + f(2) +f(3)+f(4)+f(5)+\cdots+ f(2^n)$.
Or, $\displaystyle\sum_{i=1}^{2^n} f(i)=?$
- Mon Apr 10, 2017 7:03 pm
- Forum: Junior Level
- Topic: Beginner's Marathon
- Replies: 68
- Views: 44491
Re: Beginner's Marathon
$\text{Solution to Problem 11}$ Lemma 1: $f(k)=f(2k)$ Proof: Trivial Lemma 2: $f(2k+1)=2k+1$ Proof: Trivial We will use induction. For $n=1$, it is true. Now for $g(m)=f(m+1)+f(m+2)+\cdots+f(2m)=m^2$ We will be done if we can proof $g(m+1)=f(m+2)+f(m+3)+\cdots+f(2m)+f(2m+1)+f(2m+2)=(m+1)^2$ Now,$f(m...
- Sat Apr 08, 2017 1:18 pm
- Forum: Junior Level
- Topic: geomerty
- Replies: 10
- Views: 8329
Re: geomerty
This is a Dhaka regional 2017. I solved it using scale and compass(just measured the length with my scale. Got $2017$, which was definitely the answer because this is 2017) :p