## Some unsolved problemsJanuary 3, 2010

Posted by Damien Jiang in Problem-solving.
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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.

## USAMO 1973 #1August 7, 2009

Posted by lumixedia in Problem-solving.
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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 #5August 4, 2009

Posted by lumixedia in Problem-solving.
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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, #3July 21, 2009

Posted by lumixedia in Problem-solving.
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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 #5July 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 ${\angle CBD}$.