Some unsolved problems January 3, 2010Posted by Damien Jiang in Problem-solving.
Tags: functional equation, geometry, IMO longlist, incircle, integer functional equation, Russian Olympiad
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 #1 August 7, 2009Posted by lumixedia in Problem-solving.
Tags: contest math, geometry, olympiad math, USAMO, USAMO 1973
USAMO 1973 #1. Two points, and , lie in the interior of a regular tetrahedron . Prove that angle . (more…)
USAMO 1972 #5 August 4, 2009Posted by lumixedia in Problem-solving.
Tags: contest math, geometry, olympiad math, USAMO, USAMO 1972
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…)
USAMO 1972 #2, #3 July 21, 2009Posted by lumixedia in Problem-solving.
Tags: combinatorics, contest math, geometry, olympiad math, USAMO, USAMO 1972
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…)
USAMO 2009 #5 July 19, 2009Posted by Damien Jiang in Problem-solving, Uncategorized.
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 .