BdMO 2013 Junior Problem 10

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Fatin Farhan
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BdMO 2013 Junior Problem 10

Unread post by Fatin Farhan » Wed Aug 28, 2013 11:23 am

$\Delta ABC$ এর অভ্যন্তরে O যেকোন বিন্দু। A,O;B,O এবং C,O কে যোগ করে বর্ধিত করা হলে তারা BC,AC এবং AB কে যথাক্রমে D,E এবং F বিন্দুতে ছেদ করে। AF:BF=4:3 এবং
$\Delta BOF$ ও $\Delta BOD$ এর ক্ষেত্রফল যথাক্রমে 60 ও 70 বর্গ একক।$\Delta AOF, \Delta AOE, \Delta COE$ ও$\Delta COD$ এর মধ্যে সর্বাধিক ক্ষেত্রফল বিশিষ্ট ত্রিভুজ ক্ষেত্র কোনটি এবং এর ক্ষেত্রফল কত?

There is a point O inside $\Delta ABC$. After joining A,O; B,O and C,O extend those line and they will intersect BC, AC and AB at points D, E and F respectively. AF:BF=4:3 and area of $\Delta BOF$ and $\Delta BOD$ is 60 and 70 square unit respectively. Find the triange with the largest are among $\Delta AOF, \Delta AOE, \Delta COE$ and $\Delta COD$ and write down the area of that one.
Last edited by Fatin Farhan on Wed Aug 28, 2013 4:44 pm, edited 1 time in total.
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Mursalin
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Re: BdMO 2013 Junior Problem 10

Unread post by Mursalin » Wed Aug 28, 2013 3:18 pm

@ Fatin: Are you sure this is the actual problem from the Olympiad? Because it has infinite answers. Take another triangle $A'B'C'$ where $\triangle A'B'C'$ and $\triangle ABC$ are similar but not congruent. Take a point $O'$ inside $\triangle A'B'C'$ and do the same things as before and you have yourself another triangle [with different area] that satisfies all of the conditions. Did you mean to say $[\triangle BOF]=60$ and $[\triangle BOD]=70$ because that would give you a single answer [$680$].

@ Neblina: Although Ceva's theorem is an intuitive approach to the problem Fatin posed, it is not necessary.
A much shorter approach is to make the observation that $AO=2OD$.

Let me know if you need help with anything.
Last edited by Mursalin on Wed Aug 28, 2013 4:51 pm, edited 1 time in total.
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Neblina
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Re: BdMO 2013 Junior Problem 10

Unread post by Neblina » Wed Aug 28, 2013 4:23 pm

Mursalin wrote:@ Fatin: Are you sure this is the actual problem from the Olympiad? Because it has infinite answers. Take another triangle $A'B'C'$ where $\triangle A'B'C'$ and $\triangle ABC$ are similar but not congruent. Take a point $O'$ inside $\triangle A'B'C'$ and do the same things as before and you have yourself another triangle [with different area] that satisfies all of the conditions. Did you mean to say $[\triangle BOF]=60$ and $[\triangle BOD]=70$ because that would give you a single answer [$680$].

@ Neblina: Although Ceva's theorem is an intuitive approach to the problem Fatin posed, it is not necessary.
A much shorter approach is to make the observation that $AO=2AD$.

Let me know if you need help with anything.
I think you meant AD=2AO! :P

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Mursalin
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Re: BdMO 2013 Junior Problem 10

Unread post by Mursalin » Wed Aug 28, 2013 4:53 pm

Oops!

I meant $AO=2OD$.
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Fatin Farhan
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Re: BdMO 2013 Junior Problem 10

Unread post by Fatin Farhan » Wed Aug 28, 2013 4:55 pm

Mursalin wrote:@ Fatin: Are you sure this is the actual problem from the Olympiad?
yeah.
How can i solve?
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Mursalin
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Re: BdMO 2013 Junior Problem 10

Unread post by Mursalin » Wed Aug 28, 2013 5:02 pm

Fatin Farhan wrote:$\Delta ABC$ এর অভ্যন্তরে O যেকোন বিন্দু। A,O;B,O এবং C,O কে যোগ করে বর্ধিত করা হলে তারা BC,AC এবং AB কে যথাক্রমে D,E এবং F বিন্দুতে ছেদ করে। AF:BF=4:3 এবং
$\Delta BOF$ ও $\Delta BOD$ এর ক্ষেত্রফল যথাক্রমে 60 ও 70 বর্গ একক।$\Delta AOF, \Delta AOE, \Delta COE$ ও$\Delta COD$ এর মধ্যে সর্বাধিক ক্ষেত্রফল বিশিষ্ট ত্রিভুজ ক্ষেত্র কোনটি এবং এর ক্ষেত্রফল কত?

There is a point O inside $\Delta ABC$. After joining A,O; B,O and C,O extend those line and they will intersect BC, AC and AB at points D, E and F respectively. AF:BF=4:3 and area of $\Delta BOF$ and $\Delta BOD$ is 60 and 70 square unit respectively. Find the triange with the largest are among $\Delta AOF, \Delta AOE, \Delta COE$ and $\Delta COD$ and write down the area of that one.
Well you edited the problem. Now it makes sense.

Since you changed the wording of the problem, you can check out its solution at the link Neblina posted.

I'm mentioning your previous problem so that people don't get confused about the conversations. This is what your problem looked like before the edit:

"$\Delta ABC$ এর অভ্যন্তরে O যেকোন বিন্দু। A,O;B,O এবং C,O কে যোগ করে বর্ধিত করা হলে তারা BC,AC এবং AB কে যথাক্রমে D,E এবং F বিন্দুতে ছেদ করে। AF:BF=4:3 এবং $\Delta BOF$ : $\Delta BOD=60:70$. বড় ত্রিভুজের ক্ষেত্রফল কত?"
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