Search found 98 matches

by Thamim Zahin
Mon Jun 05, 2017 10:23 pm
Forum: Geometry
Topic: IGO 2016 Medium/4
Replies: 1
Views: 322

Re: IGO 2016 Medium/4

Suppose $AB<AC$ (the solution is same if $AB>AC$) Here, $\triangle KAQ \sim \triangle KCQ \Rightarrow \frac{KA}{KC}=\frac{[KAQ]}{[KQC]}=(\frac{AQ}{QC})^2$ And, $\triangle PAB \sim \triangle PAC \Rightarrow \frac{PB}{PC}=\frac{[APB]}{[APC]}=(\frac{AB}{AC})^2$ We also have, $\triangle PQA \sim \triang...
by Thamim Zahin
Tue Apr 25, 2017 7:50 pm
Forum: Social Lounge
Topic: BDMO Forum Mafia #2
Replies: 27
Views: 1711

Re: BDMO Forum Mafia #2

10 minutes left. :(
by Thamim Zahin
Mon Apr 24, 2017 12:11 am
Forum: Social Lounge
Topic: BDMO Forum Mafia #2
Replies: 27
Views: 1711

Re: BDMO Forum Mafia #2

I have a plan: First everyone will vote themselves, so gonju(or me) will die. And from the next day, we(or just you) will continue as @thanicsamin said. Who will not do as you guys said will be killed.
by Thamim Zahin
Sat Apr 22, 2017 9:31 pm
Forum: Geometry
Topic: USA(J)MO 2017 #3
Replies: 6
Views: 420

Re: USA(J)MO 2017 #3

joydip wrote:
Proof: Let the tangent to $(ABC)$ at $A$ meet $BP$ at $J$ .Then applying pascal's theorem on hexagon $AACPBB$ we get $JF \parallel BB \parallel AC$ . So
How did you get that idea?
by Thamim Zahin
Sat Apr 22, 2017 6:17 pm
Forum: Combinatorics
Topic: USAMO 2017/4, USA(J)MO 2017/6
Replies: 0
Views: 283

USAMO 2017/4, USA(J)MO 2017/6

Let $P_1$, $P_2$, $\dots$, $P_{2n}$ be $2n$ distinct points on the unit circle $x^2+y^2=1$, other than $(1,0)$. Each point is colored either red or blue, with exactly $n$ red points and $n$ blue points. Let $R_1$, $R_2$, $\dots$, $R_n$ be any ordering of the red points. Let $B_1$ be the nearest blue...
by Thamim Zahin
Fri Apr 21, 2017 1:37 am
Forum: Geometry
Topic: USA TST 2011/1
Replies: 3
Views: 297

Re: USA TST 2011/1

Let the lines $\overleftrightarrow{FE}$ and $\overleftrightarrow{BC}$ intersect at $M$. and let the segment $\overline{FE}$ and $\overline{AD}$ intersect at $N$. And let take $\overleftrightarrow{AP}$ and $\overleftrightarrow{QR}$ meet at $P_{\infty}$. Now we know that $(M,D;B,C)$ is harmonic. Take ...
by Thamim Zahin
Fri Apr 21, 2017 12:49 am
Forum: Social Lounge
Topic: BDMO Forum Mafia #2
Replies: 27
Views: 1711

Re: BDMO Forum Mafia #2

This is actually a good idea. I am not voting myself because I am gonju(I don't have any proof but trust me)
by Thamim Zahin
Thu Apr 20, 2017 11:47 pm
Forum: Geometry
Topic: When everyone is busy solving USA(J)MO 2017,I am solving2016
Replies: 1
Views: 224

When everyone is busy solving USA(J)MO 2017,I am solving2016

The isosceles triangle $\triangle ABC$, with $AB=AC$, is inscribed in the circle $\omega$. Let $P$ be a variable point on the arc $\stackrel{\frown}{BC}$ that does not contain $A$, and let $I_B$ and $I_C$ denote the incenters of triangles $\triangle ABP$ and $\triangle ACP$, respectively. Prove that...
by Thamim Zahin
Tue Apr 18, 2017 11:45 pm
Forum: Junior Level
Topic: Beginner's Marathon
Replies: 68
Views: 4024

Re: Beginner's Marathon

$\text{Problem 16}$ There are $n$ ants on a stick of length one unit, each facing left or right. At time $t = 0$, each ant starts moving with a speed of $1$ unit per second in the direction it is facing. If an ant reaches the end of the stick, it falls off and doesn’t reappear. When two ants moving ...
by Thamim Zahin
Tue Apr 18, 2017 8:36 pm
Forum: Junior Level
Topic: Beginner's Marathon
Replies: 68
Views: 4024

Re: Beginner's Marathon

aritra barua wrote:Let us denote by $ABC$,the area of $\bigtriangleup ABC$
Use $[ABC]$ to denote the area of $\triangle ABC$. It is more natural and used frequently.