Search found 312 matches
- Wed Jan 04, 2017 10:12 pm
- Forum: Geometry
- Topic: Proving concurrency at symmedian point.
- Replies: 4
- Views: 3603
Re: Proving concurrency at symmedian point.
Let $L$ be the Lemoine Point (symmedian point) of $\triangle ABC$. We will prove that $A^{'},L,P$ collinear.(Similarly, we can prove that $B^{'},L,Q$ and $C^{'},L,R$ are collinear) Let $\triangle XYZ$ be the tangential triangke of $\triangle ABC$ with $X$ lies opposite to $A$, $Y$ lies opposite to ...
- Wed Dec 28, 2016 7:21 pm
- Forum: Combinatorics
- Topic: Iran TST 2006, D2, P3
- Replies: 0
- Views: 2069
Iran TST 2006, D2, P3
Let $G$ be a tournoment such that it's edges are colored either red or blue.
Prove that there exists a vertex of $G$ like $v$ with the property that, for every other vertex $u$ there is a mono-color directed path from $v$ to $u$.
Prove that there exists a vertex of $G$ like $v$ with the property that, for every other vertex $u$ there is a mono-color directed path from $v$ to $u$.
- Wed Dec 28, 2016 7:16 pm
- Forum: Number Theory
- Topic: IMO Shortlist 2012,N5
- Replies: 1
- Views: 2692
IMO Shortlist 2012,N5
For a nonnegative integer $n$ define $\operatorname{rad}(n)=1$ if $n=0$ or $n=1$, and $\operatorname{rad}(n)=p_1p_2\cdots p_k$ where $p_1<p_2<\cdots <p_k$ are all prime factors of $n$. Find all polynomials $f(x)$ with nonnegative integer coefficients such that $\operatorname{rad}(f(n))$ divides $\op...
- Wed Dec 28, 2016 7:11 pm
- Forum: Number Theory
- Topic: Iran 2015, TST2, D2, P2
- Replies: 0
- Views: 1922
Iran 2015, TST2, D2, P2
We call a permutation $(a_1, a_2,\cdots , a_n)$ of the set $\{ 1,2,\cdots, n\}$ "good" if for any three natural numbers $i <j <k$, $n\nmid a_i+a_k-2a_j$. Find all natural numbers $n\ge 3$ such that there exists a "good" permutation of a set $\{1,2,\cdots, n\}.$
- Fri Dec 16, 2016 4:34 pm
- Forum: Algebra
- Topic: USA Team Selection Test 2002,P1
- Replies: 0
- Views: 1980
USA Team Selection Test 2002,P1
Let $\triangle ABC$ be a triangle, and $A$, $B$, $C$ its angles. Prove that:
\[ \sin\frac{3A}{2}+\sin\frac{3B}{2}+\sin\frac{3C}{2}\leq \cos\frac{A-B}{2}+\cos\frac{B-C}{2}+\cos\frac{C-A}{2}.\]
\[ \sin\frac{3A}{2}+\sin\frac{3B}{2}+\sin\frac{3C}{2}\leq \cos\frac{A-B}{2}+\cos\frac{B-C}{2}+\cos\frac{C-A}{2}.\]
- Fri Nov 25, 2016 7:54 pm
- Forum: Combinatorics
- Topic: 'BASIC THINGS OF COMBINATORICS
- Replies: 2
- Views: 3208
Re: 'BASIC THINGS OF COMBINATORICS
To learn the very basic things, you can read this book:
http://gen.lib.rus.ec/book/index.php?md ... f813e7dc92
(Just click on the cover of the book and download will be started).
You can also continue this workshop: viewtopic.php?f=9&t=2867
http://gen.lib.rus.ec/book/index.php?md ... f813e7dc92
(Just click on the cover of the book and download will be started).
You can also continue this workshop: viewtopic.php?f=9&t=2867
- Tue Nov 22, 2016 10:18 pm
- Forum: Geometry
- Topic: APMO 2013 P5
- Replies: 3
- Views: 3676
Re: APMO 2013 P5
Already posted here:
viewtopic.php?f=15&t=2758&p=13832&hilit ... 2F5#p13832
viewtopic.php?f=15&t=2758&p=13832&hilit ... 2F5#p13832
- Sat Nov 12, 2016 3:46 pm
- Forum: Geometry
- Topic: A Beauty from Evan Chen
- Replies: 1
- Views: 2533
A Beauty from Evan Chen
Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Denote by $M$, $N$ the midpoints of $\overline{AH}$, $\overline{BC}$. Suppose the circle $\gamma$ with diameter $\overline{AH}$ meets the circumcircle of $ABC$ at $G \neq A$,and meets line $AN$ at a point $Q \neq A$. The tang...
- Sat Nov 12, 2016 3:23 pm
- Forum: Geometry
- Topic: Circle passing through the centers of two others'
- Replies: 1
- Views: 2358
Re: Circle passing through the centers of two others'
It is easy to see that $\triangle ABO_{B} \sim \triangle ACO_{C}$. So,$A$ is the center of spiral similarity which takes $BO_{B}$ to $CO_{C}$.Then from a well known result,we get that $A$ is the Miquel Point of cyclic quadrilateral $BCO_{B}O_{C}$. Now let $BC \cap O_{B}O_{C}=P$,$AH \cap BC=Q$ and $...
- Sat Nov 12, 2016 2:27 pm
- Forum: Number Theory
- Topic: Functional divisibility
- Replies: 2
- Views: 5055
Re: Functional divisibility
Let $P(x,y)$ be the assertion $f(x)+f(y) \mid (x+y)^{k}$. (1) $f(1)=1$ Proof: First assume that $f(1) \neq 1$.$P(1,1)$ implies $f(1)=2^{a}$ for some non-negative integer $a$. $P(2,2)$ implies $f(2)=2^{b}$ for some non-negative integer $b$. $P(1,2)$ implies that $f(1)+f(2) \mid 3^{k}$.If both $f(a),...