## Search found 36 matches

- Wed Jan 09, 2019 1:39 pm
- Forum: Social Lounge
- Topic: BDMO Forum Mafia #2
- Replies:
**30** - Views:
**26920**

- Sun Sep 02, 2018 1:13 pm
- Forum: Social Lounge
- Topic: Chat thread
- Replies:
**53** - Views:
**43259**

### Re: Chat thread

♪＼(*＾▽＾*)／＼(*＾▽＾*)／

- Tue May 15, 2018 8:25 pm
- Forum: Social Lounge
- Topic: Chat thread
- Replies:
**53** - Views:
**43259**

### Re: Chat thread

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- Thu Apr 19, 2018 10:25 pm
- Forum: Geometry
- Topic: EGMO 2018 P5
- Replies:
**3** - Views:
**9742**

### Re: EGMO 2018 P5

Let $N$ be the intersection point of $(ABC)$ and the angle bisector of $\angle{ACB}$ other than $C$ Suppose $\Omega$ touches $\tau$ at $L$ and $AB$ at $M$ Let $M'=NL \cap AB$ and $XY$ be the common tangent of $\tau$ and $\Omega$ Define $K=XY \cap AB$ $KM=KL$ $\measuredangle KLM' = \measuredangle KLN...

- Thu Apr 19, 2018 10:21 pm
- Forum: Geometry
- Topic: EGMO 2018 P5
- Replies:
**3** - Views:
**9742**

### EGMO 2018 P5

Let $\tau$ be the circumcircle of triangle $ABC$. A circle $\Omega$ is tangent to the line segment $AB$ and is tangent to $\tau$ at a point lying on the same side of the line $AB$ as $C$. The angle bisector of $\angle{BCA}$ intersects $\Omega$ at two different points $P$ and $Q$. Prove that, $\angle...

- Wed Apr 18, 2018 6:31 pm
- Forum: Secondary Level
- Topic: Easy Projective Geo
- Replies:
**3** - Views:
**4260**

### Re: Easy Projective Geo

We get $ACDE$ as an isosceles trapezoid Angle chasing gives us that $\angle{ABE}=\angle{DBC}$ Let $Q$ be the intersection point of $AC$ and $BD$. Again, $(A,C;Q,P)$ is a harmonic bundle $\frac{AQ}{CQ}=\frac{AP}{CP}$ Now it is enough to show that $BD$ is a symmedian Which means to show, $\frac{AB^2}{...

- Sun Apr 15, 2018 10:05 pm
- Forum: National Math Camp
- Topic: 2018 NT exam P2
- Replies:
**0** - Views:
**3625**

### 2018 NT exam P2

Let $a_1, a_2, ....., a_n$ be a sequence of real numbers such that $a_1+a_2+......a_n=0$ and define $b_i=a_1+a_2+......a_i$ for $$1\leq i \leq n$$. Suppose that $$b_i(a_{j+1}-a_{i+1})\geq 0$$ for all $$1\leq i \leq j \leq n-1$$. Show that max |$a_l$| $\geq$ max |$b_m$| $(1 \leq l \leq n)$ and $(1 \l...

- Thu Apr 12, 2018 3:43 pm
- Forum: Number Theory
- Topic: 2018 BDMO NT exam P4
- Replies:
**2** - Views:
**4439**

### 2018 BDMO NT exam P4

Let $n$ be an even positive integer. A sequence of $n$ real numbers is called complete if for every integer $m$ with $$1 \leq m \leq n$$ either the sum of the first $m$ terms or the sum of the last $m$ terms is integral. Determine the minimum number of the integers in a complete sequence of $n$ numb...

- Mon Apr 09, 2018 5:21 pm
- Forum: Geometry
- Topic: BAYMAX is cyclic!
- Replies:
**1** - Views:
**7210**

### Re: BAYMAX is cyclic!

Let $N$ and $R$ be the midpoints of $AB$ and $BC$ resp. Then we get $NRQP$ is a parallelogram. Again, $AA_Y.AB=AP.AC$ means, $AA_Y.AN=AP.AQ$ It gives $NA_YPQ$ cyclic. Similarly, $QMRA_X$ is cyclic. We are working with directed angle $\angle{APN}=-\angle{AA_YM}$ Again, $\angle{APN}=\angle{AQR}=\angle...

- Mon Apr 09, 2018 5:06 pm
- Forum: Geometry
- Topic: BAYMAX is cyclic!
- Replies:
**1** - Views:
**7210**

### BAYMAX is cyclic!

Let $M$ be the midpoint of the side $AC$ of triangle $ABC$. Let $P$ on $AM$ and $Q$ on $CM$ be such that $PQ=AC/2$. Let $(ABQ)$ intersect with $BC$ at $A_X$ other than $B$. $(BCP)$ intersects with $BA$ at $A_Y$ other than $B$. Prove that $BA_YMA_X$ is cyclic