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BdMO Online Forum • Search

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Re: Beginner's Marathon

An easy problem:$\text{Problem 20}$ The fractions $\frac{7x+1}{2}$, $\frac{7x+2}{3}$,......., $\frac{7x+2016}{2017}$ are irreducible.Find all possible values of $x$ such that $x$ is less than or equal to $300$.This Problem was recommended by Zawad bhai in a mock test at CMC. Here's a hint for you g...
by Zawadx
Tue Jun 13, 2017 4:38 pm
 
Forum: Junior Level
Topic: Beginner's Marathon
Replies: 68
Views: 2802

Re: MPMS Problem Solving Marathon

aritra barua wrote:For problem 1,if we substitute $a^2$=$4k$ where $k$ is an integer,we can easily find that $p$ |$a^2$+$1$.


For this you need to have $4k$ a perfect square, which isn't true for all $k$.
by Zawadx
Mon May 29, 2017 5:02 pm
 
Forum: News / Announcements
Topic: MPMS Problem Solving Marathon
Replies: 8
Views: 428

Re: FE: Brahmagupta-Fibonacci identity!!!

No one posted any solution. Sillies. Let $P(x,y,z,t)$ denote the statement that $[f(x)+f(y)][f(z)+f(t)]=f(xz+yt)+f(xt-yz) $ Now, $P(0,0,0,0) \Rightarrow 2f(0)(2f(0) - 1) = 0$ So we have $f(0)=\dfrac{1}{2}$ or $f(0)=0$ Suppose $f(0)=\dfrac{1}{2}$ . Then, $P(x,0,0,0) \Rightarrow f(x)=\dfrac{1}{2}$ We ...
by Zawadx
Fri May 12, 2017 1:44 am
 
Forum: Algebra
Topic: FE: Brahmagupta-Fibonacci identity!!!
Replies: 1
Views: 172

Re: Geometry Marathon : Season 3

$\text{Problem 41}$

Let $ABC$ be a triangle, and $I$ the incenter, $M$ midpoint of $ BC $, $ D $ the touch point of incircle and $ BC $. Prove that perpendiculars from $M, D, A $ to $AI, IM, BC $ respectively are concurrent.
by Zawadx
Mon May 01, 2017 3:10 am
 
Forum: Geometry
Topic: Geometry Marathon : Season 3
Replies: 90
Views: 3232

Re: Geometry Marathon : Season 3

$\text{Solution to Problem 40:}$ Since $N$ is the Nagel point, we can easily prove that $BE=CD$ i.e. $BM=CT$. $P$ is the center of spiral similarity which sends $BE$ to $CD$ thus it also sends $M$ to $T$. Since $BE=CD$, we have the dilation factor at $1$ thus $PM=PT$. Now since a spiral similarity c...
by Zawadx
Mon May 01, 2017 2:35 am
 
Forum: Geometry
Topic: Geometry Marathon : Season 3
Replies: 90
Views: 3232

Re: BDMO Forum Mafia #2

Nobody died.

Day 2 begins now and will end on 28th April, 8 pm.
by Zawadx
Wed Apr 26, 2017 8:22 pm
 
Forum: Social Lounge
Topic: BDMO Forum Mafia #2
Replies: 27
Views: 1113

Re: BDMO Forum Mafia #2

Nobody died, yay!

Night 1 starts now, will end at 26th April 8 pm.

I hope the mafia remember who they are, cause I totally forgot :roll:
by Zawadx
Tue Apr 25, 2017 8:05 pm
 
Forum: Social Lounge
Topic: BDMO Forum Mafia #2
Replies: 27
Views: 1113

Re: BDMO Forum Mafia #2

Since this is dying, gonna fix a final deadline.

Day 1 will end on 25th, 8 pm. Discuss, do stuff, act!
by Zawadx
Mon Apr 24, 2017 8:06 pm
 
Forum: Social Lounge
Topic: BDMO Forum Mafia #2
Replies: 27
Views: 1113

Re: USA(J)MO 2017 #3

We use barycentric coordinates. Let $P\equiv (p:q:r)$. Now, we know that $pq+qr+rp=0$ [The equation of circumcircle for equilateral triangles]. Now, $D\equiv (0:q:r), E\equiv (p:0:r), F\equiv (p:q:0)$. So, the area of $\triangle DEF$ divided by the area of $\triangle ABC$ is: $$\dfrac{1}{(p+q)(q+r)...
by Zawadx
Sat Apr 22, 2017 6:48 pm
 
Forum: Geometry
Topic: USA(J)MO 2017 #3
Replies: 6
Views: 293

Re: BDMO Forum Mafia #2

Bump bump bumpity bump.

How much time should I extend for?
by Zawadx
Sat Apr 22, 2017 8:27 am
 
Forum: Social Lounge
Topic: BDMO Forum Mafia #2
Replies: 27
Views: 1113
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