geometry | Delta Epsilons

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

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

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.
Tags: contest math, geometry, olympiad math, USAMO, USAMO 1972
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.
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 ${ABCD}$ 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.
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 ${ABCD}$ , 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 ${\angle CBD}$ .

(more…)

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Delta Epsilons

Some unsolved problems January 3, 2010

USAMO 1973 #1 August 7, 2009

USAMO 1972 #5 August 4, 2009

USAMO 1972 #2, #3 July 21, 2009

USAMO 2009 #5 July 19, 2009

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