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...

- Mon Jun 05, 2017 10:23 pm
- Forum: Geometry
- Topic: IGO 2016 Medium/4
- Replies:
**1** - Views:
**194**

10 minutes left.

- Tue Apr 25, 2017 7:50 pm
- Forum: Social Lounge
- Topic: BDMO Forum Mafia #2
- Replies:
**27** - Views:
**1003**

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.

- Mon Apr 24, 2017 12:11 am
- Forum: Social Lounge
- Topic: BDMO Forum Mafia #2
- Replies:
**27** - Views:
**1003**

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?

- Sat Apr 22, 2017 9:31 pm
- Forum: Geometry
- Topic: USA(J)MO 2017 #3
- Replies:
**6** - Views:
**267**

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...

- Sat Apr 22, 2017 6:17 pm
- Forum: Combinatorics
- Topic: USAMO 2017/4, USA(J)MO 2017/6
- Replies:
**0** - Views:
**142**

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 ...

- Fri Apr 21, 2017 1:37 am
- Forum: Geometry
- Topic: USA TST 2011/1
- Replies:
**3** - Views:
**180**

This is actually a good idea. I am not voting myself because I am gonju(I don't have any proof but trust me)

- Fri Apr 21, 2017 12:49 am
- Forum: Social Lounge
- Topic: BDMO Forum Mafia #2
- Replies:
**27** - Views:
**1003**

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...

- 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:
**129**

$\text{Problem 16}$ 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 ...

- Tue Apr 18, 2017 11:45 pm
- Forum: Junior Level
- Topic: Beginner's Marathon
- Replies:
**68** - Views:
**2560**

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.

- Tue Apr 18, 2017 8:36 pm
- Forum: Junior Level
- Topic: Beginner's Marathon
- Replies:
**68** - Views:
**2560**