Darn USAMOs are hard

by djmathman, Jan 22, 2013, 1:12 AM

so I have to resort to solving simpler versions of the problems.
(Almost) USAMO 2001/4 wrote:
Let $P$ be a point in the interior of triangle $ABC$ such that the segments $PA$, $PB$, and $PC$ are the sides of an obtuse triangle. Assume that in this triangle the obtuse angle opposes the side congruent to $PA$. Prove that $\angle BAC$ is acute.

and then the problem becomes one I can actually solve
This post has been edited 1 time. Last edited by djmathman, Jan 22, 2013, 1:12 AM

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If I remember correctly you can coordinate bash this pretty easily.

But yeah there's a big jump from 1/4s to 2/5s that takes some time to get across...

by NewAlbionAcademy, Jan 22, 2013, 1:48 AM

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Well the thing I'm having trouble with in the original USAMO problem is the case where $P$ lies outside the triangle. I keep trying to use the relation $PA^2>PB^2+PC^2$ but I can't seem to use it for any good.

by djmathman, Jan 22, 2013, 1:53 AM

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If you set A to be the origin and B to be on the x axis, then if you assume to the contrary then you can wlog that C is in the 2nd quadrant (or the y axis...but that doesn't change anything).

Then I think coordinates kill it.
This post has been edited 1 time. Last edited by NewAlbionAcademy, Jan 23, 2013, 3:26 AM

by NewAlbionAcademy, Jan 23, 2013, 3:26 AM

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  • dj so orz :omighty:

    by Yiyj1, Mar 29, 2025, 1:42 AM

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  • orz $$\,$$

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  • waiting for a recap of your amc proposals for this year :D

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  • Do you have a link to your main blog that you started after graduating from high school, I couldn't find it. @dj I met you IRL at Awesome Math summer Program several years ago.

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