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January 19th, 2011, 05:56 AM  #1 
Senior Member Joined: Oct 2010 From: Vietnam Posts: 226 Thanks: 0  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 nonnegative 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 Xset to be a set of n blue points having distinct xcoordinates, and a Yset to be a set of n blue points having distinct ycoordinates. Prove that the number of Xsets = the number of Ysets. 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. 
January 19th, 2011, 10:26 AM  #3 
Senior Member Joined: Nov 2010 Posts: 288 Thanks: 1  Re: Some questions about International Mathematical Olympiad
Where do you get these solutions from?

January 20th, 2011, 05:38 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,965 Thanks: 2214 
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?

January 20th, 2011, 06:35 AM  #5  
Senior Member Joined: Nov 2010 Posts: 288 Thanks: 1  Re: Quote:
 
January 20th, 2011, 07:25 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,965 Thanks: 2214 
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. 
January 20th, 2011, 08:33 AM  #7 
Senior Member Joined: Nov 2010 Posts: 288 Thanks: 1  Re: Some questions about International Mathematical Olympiad
thanks alot

January 21st, 2011, 11:25 PM  #8 
Senior Member Joined: Oct 2010 From: Vietnam Posts: 226 Thanks: 0  Re: Some questions about International Mathematical Olympiad
btw,what is the smallest number of ages can we participate IMO?

January 22nd, 2011, 12:05 PM  #9 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,962 Thanks: 1147 Math Focus: Elementary mathematics and beyond  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.

January 23rd, 2011, 04:36 AM  #10  
Senior Member Joined: Oct 2010 From: Vietnam Posts: 226 Thanks: 0  Re: Some questions about International Mathematical Olympiad Quote:
 

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