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Some unsolved problems
*January 3, 2010*

*Posted by Damien Jiang in Problem-solving.*

Tags: functional equation, geometry, IMO longlist, incircle, integer functional equation, Russian Olympiad

9 comments

Tags: functional equation, geometry, IMO longlist, incircle, integer functional equation, Russian Olympiad

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.*

Tags: contest math, geometry, olympiad math, USAMO, USAMO 1973

2 comments

Tags: contest math, geometry, olympiad math, USAMO, USAMO 1973

2 comments

**USAMO 1973 #1.** Two points, and , lie in the interior of a regular tetrahedron . Prove that angle . (more…)

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USAMO 1972 #5
*August 4, 2009*

*Posted by lumixedia in Problem-solving.*

Tags: contest math, geometry, olympiad math, USAMO, USAMO 1972

2 comments

Tags: contest math, geometry, olympiad math, USAMO, USAMO 1972

2 comments

**USAMO 1972 #5.** A given convex pentagon has the property that the area of each of the five triangles , , , , 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…)

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USAMO 1972 #2, #3
*July 21, 2009*

*Posted by lumixedia in Problem-solving.*

Tags: combinatorics, contest math, geometry, olympiad math, USAMO, USAMO 1972

3 comments

Tags: combinatorics, contest math, geometry, olympiad math, USAMO, USAMO 1972

3 comments

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

**USAMO 1972 #2.** A given tetrahedron is isosceles, that is , , . Show that the faces of the tetrahedron are acute-angled triangles. (more…)

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USAMO 2009 #5
*July 19, 2009*

*Posted by Damien Jiang in Problem-solving, Uncategorized.*

Tags: geometry, olympiad math

1 comment so far

Tags: geometry, olympiad math

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 , with , is inscribed in circle and point lies inside triangle . Rays and meet again at points and , respectively. Let the line through parallel to intersects and at points and , respectively. Prove that quadrilateral is cyclic if and only if bisects .