National BDMO 2017 : Junior 8

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dshasan
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National BDMO 2017 : Junior 8

Unread post by dshasan » Fri Feb 10, 2017 9:56 pm

In $\bigtriangleup ABC$, the perpendicular bisector of $AB$ and $AC$ meet at $O$. $AO$ meets $BC$ at $D$. Now, $OD$ = $BD$ = $\dfrac {1}{3}BC$. Find the angles of $\bigtriangleup ABC$.
The study of mathematics, like the Nile, begins in minuteness but ends in magnificence.

- Charles Caleb Colton

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Thamim Zahin
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Re: National BDMO 2017 : Junior 8

Unread post by Thamim Zahin » Sat Feb 11, 2017 11:55 pm

Are there anyone who solve this in the exam time? Did you @dshasan?
I think we judge talent wrong. What do we see as talent? I think I have made the same mistake myself. We judge talent by the trophies on their showcases, the flamboyance the supremacy. We don't see things like determination, courage, discipline, temperament.

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Thamim Zahin
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Re: National BDMO 2017 : Junior 8

Unread post by Thamim Zahin » Sun Feb 12, 2017 9:19 pm

We know that O is the circumcenter. So, $OA=OB=OC$. And $DB=DO$. So, $ \angle DBO=\angle DOB=a$.

Take D' such that $D'C= \frac {1}{3} BC$. So, $DO=DB=DD'=D'C$.

$OB=OC$. So, $\angle OBC= \angle OCB=a$. So by $SAS$ congruence $\triangle OBD \cong \triangle OCD'$.
So, $OD=OD'$, And, $\triangle ODD'$ is equilateral. And $\angle ODD'=2a$. So, $a=30^o$. The rest are calculation.

So, $\angle BOA = 180^o-a=150^o$. So, $\angle OAB=\angle OBA= 15^o$.

And now, $\angle COA = 180^o-3a=90^o$. So, $\angle OAC=\angle OCA= 45^o$.

So, $\angle ABC=30^o+15^o= 45^o$. $\angle BCA= 30^o+45^o=75^o$.

And, $\angle CAB= 45^o+15^o= 60^o$.

Answer $(\angle A,\angle B,\angle C)=(60^o,45^o,75^o)$
I think we judge talent wrong. What do we see as talent? I think I have made the same mistake myself. We judge talent by the trophies on their showcases, the flamboyance the supremacy. We don't see things like determination, courage, discipline, temperament.

dshasan
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Re: National BDMO 2017 : Junior 8

Unread post by dshasan » Sun Feb 12, 2017 9:32 pm

Take point E on $BC$ such that $BD = DE = EC$. Join $O, E$. Now, by $SAS$ theorem, $\bigtriangleup OCE$ is congruent to $\bigtriangleup OBD$. SO, $OD = OE =DE$ and $\bigtriangleup ODE$ is an equilateral triangle. So, $\angle OBD = \angle BOD = 30$. Again, $\angle BAD + \angle ABO + \angle OBD = 60 \Rightarrow \angle BAD + \angle ABO = 30$. Or, $\angle BAD = \angle ABO = 15$. SO, $\angle B = 45$.
Same way, we can find that $\angle C = 75$ and $\angle A = 60$.
@Thamim, yes, I solved this during exam hour. :)
The study of mathematics, like the Nile, begins in minuteness but ends in magnificence.

- Charles Caleb Colton

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Thamim Zahin
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Re: National BDMO 2017 : Junior 8

Unread post by Thamim Zahin » Mon Feb 13, 2017 4:54 pm

I am actually kicking me for not trying 8,9 in the exam. They both are easy.
I think we judge talent wrong. What do we see as talent? I think I have made the same mistake myself. We judge talent by the trophies on their showcases, the flamboyance the supremacy. We don't see things like determination, courage, discipline, temperament.

Mamnoon Siam
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Re: National BDMO 2017 : Junior 8

Unread post by Mamnoon Siam » Mon Feb 13, 2017 7:28 pm

Don't kick yourself. You have 1 year more to be the COC or COO. :)

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ahmedittihad
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Re: National BDMO 2017 : Junior 8

Unread post by ahmedittihad » Mon Feb 13, 2017 11:47 pm

How does one actually kick himself?
Frankly, my dear, I don't give a damn.

aritra barua
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Re: National BDMO 2017 : Junior 8

Unread post by aritra barua » Thu Feb 16, 2017 8:51 pm

You are right,Thamim...I also feel like kicking myself for not paying much attention to this problem as I was working on other problems...or else I could have attained a bit more than being 2nd runners up......Now,This problem does need only a bit of algebraic calculations..We may let angle BAO to be y,then we may ensure that both AOB And AOC are isosceles triangles.Then by angle chasing,find angle BOD=angle OBD=angle OCD.Then by the rest angle calculations,derive an equation which follows 3y+5y+4y=180,find y =15..That fulfills our requirements of 60-75-45 angled triangle...

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ahmedittihad
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Re: National BDMO 2017 : Junior 8

Unread post by ahmedittihad » Sat Feb 18, 2017 7:32 pm

ahmedittihad wrote:How does one actually kick himself?
This is a valid question. :D :D :D
Frankly, my dear, I don't give a damn.

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