Why $O_{1}O_{2} \parallel MN$?tanmoy wrote:Because then $OF \perp O_{1}O_{2}$ and as $O_{1}O_{2} \parallel MN$,so we will be done
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- Wed Aug 17, 2016 2:36 am
- Forum: Geometry
- Topic: A Problem of Romanian TST
- Replies: 10
- Views: 8019
Re: A Problem of Romanian TST
- Tue Aug 16, 2016 10:21 pm
- Forum: Geometry
- Topic: A Problem of Romanian TST
- Replies: 10
- Views: 8019
Re: A Problem of Romanian TST
So,our goal is to show that $F$ also lies on $l$ i.e. if $ME \cap AF=P,NG \cap BF=Q$,we have to prove that $FP \cdot FA=FQ \cdot FB$. That' not true actually. But the problem is very easy from here. Let $\omega=\bigodot (M,MA),\lambda=\bigodot(N,NB)$ Let $\omega\cap AF=A,X$ and $\lambda\cap BF=B,Y$...
- Thu Aug 11, 2016 11:51 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 115664
Re: IMO Marathon
$\textbf{Problem} \text{ }\boxed{41}$ FInd all primes $p$ for which there exists $n\in \mathbb{N}$ so that
$$
p|n^{n+1}-(n+1)^n
$$
[Harder version: Replace $p$ with a general integer $m$]
$$
p|n^{n+1}-(n+1)^n
$$
[Harder version: Replace $p$ with a general integer $m$]
- Tue Aug 09, 2016 9:48 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 115664
Re: IMO Marathon
If $a_1,a_2,a_3,a_4,a_5$ are positive reals bounded below and above by $p$ and $q$ respectively $\left(0<p\le q\right)$, then prove that \[\left(a_1+a_2+a_3+a_4+a_5\right)\left(\dfrac{1}{a_1}+\dfrac{1}{a_2}+\dfrac{1}{a_3}+\dfrac{1}{a_4}+\dfrac{1}{a_5}\right)\le 25+6\left(\sqrt{\dfrac{p}{q}}-\sqrt{\...
- Tue Aug 09, 2016 4:27 pm
- Forum: Combinatorics
- Topic: even odd even odd
- Replies: 7
- Views: 5820
Re: even odd even odd
Solution: I think my notation isn't clear. I meant when considering only values, $x_i = y_i = i$. But when I say $x_i$, I am refering to the row no. of the $i$'th row, not the column no. of the $i$'th column. Now let $\sum_{(x_i,y_j)\in S}x_i+y_j = N$ We'll show $N$ is even. First set $N = 0.$ Then...
- Tue Aug 09, 2016 10:55 am
- Forum: Combinatorics
- Topic: even odd even odd
- Replies: 7
- Views: 5820
Re: even odd even odd
I don't see how to use this hint. Write your whole solution please.Nayeemul Islam Swad wrote:Hint:
- Mon Aug 08, 2016 1:10 am
- Forum: Combinatorics
- Topic: even odd even odd
- Replies: 7
- Views: 5820
Re: even odd even odd
No, as $0$ is an even number.Golam Musabbir Joy wrote:Can any row or column be empty?
- Sun Aug 07, 2016 8:59 pm
- Forum: Combinatorics
- Topic: Maximizing edges
- Replies: 2
- Views: 2714
Re: Maximizing edges
This is just a special case of Turan's theorem.Thanic Nur Samin wrote:Let there be $n$ points in a space. Some edges are connecting them, making a graph. Maximize the number of edges so that there is no tetrahedron in the graph.
https://en.m.wikipedia.org/wiki/Tur%C3%A1n's_theorem
- Sun Aug 07, 2016 7:40 pm
- Forum: Social Lounge
- Topic: Chat thread
- Replies: 53
- Views: 82537
Re: Chat thread
I think we can revive the IMO marathon too.
- Sun Aug 07, 2016 12:17 pm
- Forum: Number Theory
- Topic: IMO Shortlist 2012 N1
- Replies: 7
- Views: 5523
Re: IMO Shortlist 2012 N1
There is no condition saying $kx^2 \in A$. You have to prove it. (Though the proof is very obvious).