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January 19th, 2011, 05:56 AM   #1
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Some questions about International Mathematical Olympiad

1. Let n be a positive number. Let T be the set of points (x,y) in the plane where x,y are non-negative integers and x+y<n. Each point of T is colored red or blue. If a point (x,y) is red, then so are all points
(x,y) of T with both x?x and y?y. Define an X-set to be a set of n blue points having distinct x-coordinates, and a Y-set to be a set of n blue points having distinct y-coordinates. Prove that the number of X-sets = the number of Y-sets.

2. Let BC be a diameter of circle W with center O. Let A be a point of circle W such that 0 ? ?AOB ? 120. Let D be the midpoint of arc AB not containing C. Line l passes through O and is parallel to line AD. Line l intersects line AC at J. The perpendicular bisector of segment OA intersects circle W at E and F. Prove that J is the incenter of triangle CEF.
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January 19th, 2011, 06:20 AM   #2
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Solutions are given here.
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January 19th, 2011, 10:26 AM   #3
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Re: Some questions about International Mathematical Olympiad

Where do you get these solutions from?
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January 20th, 2011, 05:38 AM   #4
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Are you asking how one can tackle and solve these particular problems under exam conditions, or how one can find most IMO problems and their solutions on the internet?
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January 20th, 2011, 06:35 AM   #5
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Re:

Quote:
Originally Posted by skipjack
Are you asking how one can tackle and solve these particular problems under exam conditions, or how one can find most IMO problems and their solutions on the internet?
actually both
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January 20th, 2011, 07:25 AM   #6
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Mathematical induction is worth trying if the problem looks as though it might be tackled using it. I hadn't tried the geometry problem before seeing the solution, but it's usually helpful to be aware of a range of theorems.

Finding the problems and solutions on the internet can usually be accomplished by use of Google.
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January 20th, 2011, 08:33 AM   #7
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Re: Some questions about International Mathematical Olympiad

thanks alot
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January 21st, 2011, 11:25 PM   #8
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Re: Some questions about International Mathematical Olympiad

btw,what is the smallest number of ages can we participate IMO?
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January 22nd, 2011, 12:05 PM   #9
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Re: Some questions about International Mathematical Olympiad

I've looked at a page here and it says ". . . for High School students . . .". I didn't see anything specific but I didn't look too hard.
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January 23rd, 2011, 04:36 AM   #10
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Re: Some questions about International Mathematical Olympiad

Quote:
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". . . for High School students . . ."
High School?I thought it's for grade 8!
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