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Some unsolved problems January 3, 2010

Posted by Damien Jiang in Problem-solving.
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9 comments

Happy New Year!

Since we have been too lazy to post lately (and the so-not-lazy Akhil posts mostly elsewhere now), I’m going to post some problems that I probably should be able to solve, but haven’t.

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USAMO 1973 #1 August 7, 2009

Posted by lumixedia in Problem-solving.
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2 comments

USAMO 1973 #1. Two points, {P} and {Q}, lie in the interior of a regular tetrahedron {ABCD}. Prove that angle {PAQ<60^{\circ}}. (more…)

USAMO 1972 #5 August 4, 2009

Posted by lumixedia in Problem-solving.
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2 comments

USAMO 1972 #5. A given convex pentagon {ABCDE} has the property that the area of each of the five triangles {ABC}, {BCD}, {CDE}, {DEA}, {EAB} is unity. Show that every non-congruent pentagon with the above property has the same area, and that, furthermore, there are an infinite number of such non-congruent pentagons. (more…)

USAMO 1972 #2, #3 July 21, 2009

Posted by lumixedia in Problem-solving.
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3 comments

I think I might as well just start going through the USAMOs in chronological/numerical order.

USAMO 1972 #2. A given tetrahedron {ABCD} is isosceles, that is {AB=CD}, {AC=BD}, {AD=BC}. Show that the faces of the tetrahedron are acute-angled triangles. (more…)

USAMO 2009 #5 July 19, 2009

Posted by Damien Jiang in Problem-solving, Uncategorized.
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1 comment so far

I like Olympiad geometry. Therefore, I will give my solution to this year’s USAMO #5; I was rather happy with my solution.

5. Trapezoid {ABCD}, with {\overline{AB}||\overline{CD}}, is inscribed in circle {\omega} and point {G} lies inside triangle {BCD}. Rays {AG} and {BG} meet {\omega} again at points {P} and {Q}, respectively. Let the line through {G} parallel to {\overline{AB}} intersects {\overline{BD}} and {\overline{BC}} at points {R} and {S}, respectively. Prove that quadrilateral {PQRS} is cyclic if and only if {\overline{BG}} bisects {\angle CBD}.

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