Search found 110 matches

by Tahmid
Mon Mar 02, 2015 12:44 am
Forum: Number Theory
Topic: Some GCD Problems
Replies: 6
Views: 2041

Re: Some GCD Problems

tanmoy wrote:.$a^{2^{n}}+1$ is a divisor of $a^{2^{m}}-1$.
how ? :?: :?:
$a=3$ ; $n=2$ ; $m=3$ breaks it .
by Tahmid
Sun Mar 01, 2015 3:41 pm
Forum: Number Theory
Topic: Some GCD Problems
Replies: 6
Views: 2041

Re: Some GCD Problems

just apply $euclidean$ $algorithm$. nothing else.
by Tahmid
Sat Feb 28, 2015 2:39 am
Forum: Geometry
Topic: Bulgaria 1996
Replies: 4
Views: 1841

Re: Bulgaria 1996

let , $k_{1},k_{2}$ touches $k$ at points $X,Y$ respectively. and $l\cap AB=Z$ so, $AX,BY,CZ$ are concurrent at the orthocentre of triangle $ABC$ by ceva's theorem we have $\frac{AZ}{ZB}=\frac{r_{1}}{r_{2}}$ now let, $AO_{1}\cap BC=R$ ; $BO_{2}\cap AC=S$ then , $\frac{CS}{SA}\cdot \frac{AZ}{ZB}\cdot...
by Tahmid
Sat Feb 28, 2015 1:21 am
Forum: Geometry
Topic: Determine the angles
Replies: 1
Views: 980

Re: Determine the angles

$\angle A=90$
$\angle B=22.5$
$\angle C=67.5$
by Tahmid
Thu Feb 26, 2015 7:44 pm
Forum: Geometry
Topic: Sine and Cosine
Replies: 2
Views: 1324

Re: Sine and Cosine

very cool :P
need to prove $sin \angle C=2sin \angle APC cos \angle APC$

find the value of $cos \angle APC$ by cosine rule in triangle $APC$
then use stewart's theorem to find $AP^{2}$ .

plugging them into $2sin \angle APC cos \angle APC$ and just manipulate .
by Tahmid
Wed Feb 25, 2015 11:24 pm
Forum: Geometry
Topic: USAMO 2009/5
Replies: 4
Views: 3065

Re: USAMO 2009/5

i have solved this . but my solution is too large :?
main part of my solution is to prove $DX$=$CY$ where $X=QR\cap w ; Y=PS\cap w$
by Tahmid
Wed Feb 25, 2015 10:55 am
Forum: Geometry
Topic: USAMO 1999/6
Replies: 2
Views: 1311

Re: USAMO 1999/6

tanmoy ,
did you solve this ? ...
if not , then try to show that $E$ is the touch point of the excircle of triangle $ADC$ with side $DC$
by Tahmid
Tue Feb 24, 2015 8:07 pm
Forum: Geometry
Topic: Balkan MO 2005
Replies: 3
Views: 1525

Re: Balkan MO 2005

well, i have proved the first part in differrent way . here it is , let $BX\cap AC={X}'$ and the incircle touches $BC$ at $F$ now , apply menelus's $ \frac{AE}{E{X}'}\frac{{X}'X}{XB}\frac{BD}{DA}=1 \Leftrightarrow \frac{{X}'X}{XB}=\frac{E{X}'}{BD}=\frac{E{X}'}{BF}$ as , $\frac{{X}'X}{XB}=\frac{CX}{C...
by Tahmid
Tue Feb 24, 2015 12:08 pm
Forum: Geometry
Topic: Balkan MO 2005
Replies: 3
Views: 1525

Balkan MO 2005

Let $ABC$ be an acute angled triangle whose inscribed circle touches $AB$ and $AC$ at $D$ and $E$ respectively . Let $X$ and $Y$ be the point of intersection of the bisectors of the angles $\angle ACB$ and $\angle ABC$ eith the line $DE$ and let $Z$ be the midpoint of $BC$. Prove that the triangle $...
by Tahmid
Mon Feb 23, 2015 10:42 pm
Forum: Number Theory
Topic: USAJMO 2011/1
Replies: 2
Views: 1264

Re: USAJMO 2011/1

if $n=1$ ; then $2^{n}+12^{n}+2011^{n}$ is a perfect square .

take $mod 3$ when $n=2m$ for all $m\in \mathbb{N}$
take $mod 4$ when $n=2m+1$ for all $m\in \mathbb{N}$
in each case a contradiction creates.

so $n=1$ is only the solution .