Problem 4:
Triangle $ABC$ is acute and $M$ is its circumcenter. Determine what point $P$ inside the triangle satisfy \[1\le \frac {\angle APB}{\angle ACB} \le 2,\ 1\le \frac{\angle BPC}{\angle BAC}\le 2,\ 1\le \frac {\angle CPA}{\angle CBA} \le 2\]
BdMO National Higher Secondary 2009/4
BdMO National Higher Secondary 2009/4
"Inspiration is needed in geometry, just as much as in poetry."  Aleksandr Pushkin
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 Joined: Tue Sep 27, 2011 12:18 am
Re: BdMO National Higher Secondary 2009/4
AM, BM ,CM are the locus of p that satisfies the given inequality. the proof is very trivial.....
Re: BdMO National Higher Secondary 2009/4
I'm not sure about it.But I'm trying to solve it.If it is wrong plz send the write answer.
I think the point p & M is same.because \[\angle APB> AMB> ACB.\] When P & M are different point.
notice that,\[ \angle AMB= 2 \left (\angle ACB \right )\] in similar fashion we can prove the other eqation also.
So,M & P are same point.
I think the point p & M is same.because \[\angle APB> AMB> ACB.\] When P & M are different point.
notice that,\[ \angle AMB= 2 \left (\angle ACB \right )\] in similar fashion we can prove the other eqation also.
So,M & P are same point.
 nafistiham
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Re: BdMO National Higher Secondary 2009/4
I also did the same solution.and, i believe it is OK.we can say it like this, if $P$ is any other point but $M$,at least one of the equation can not be fulfilled
\[\sum_{k=0}^{n1}e^{\frac{2 \pi i k}{n}}=0\]
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