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        riddles >> easy >> Angles in GP
        
(Message started by: ronnodas on Mar 2^nd, 2009, 7:21pm)

Title: Angles in GP
Post by ronnodas on Mar 2^nd, 2009, 7:21pm

All points on a plane are colored with three colors. Prove that there exist an infinite number of triangles on the plane with all vertices the same color, that are either isosceles or have their angles in geometric progression.

Prove the statement when geometric progression is replaced by arithmetic progression and degenerate triangles are allowed.

Title: Re: Angles in GP
Post by ronnodas on Mar 8^th, 2009, 11:59pm

No takers?

The problems are variations of one given in this year's Indian Math Olympiad.

Title: Re: Angles in GP
Post by towr on Mar 9^th, 2009, 4:03am

Here (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_easy;action=display;num=1083267737)'s a thread that deal with two-colored plane. In that case it applies to any triangle.

Title: Re: Angles in GP
Post by towr on Mar 9^th, 2009, 8:45am

We can use the idea used at this site (http://skepticsplay.blogspot.com/2008/09/monochromatic-triangles-in-multi.html).
It may be interesting to do it without using [hide]Van der Waerden's Theorem[/hide] (http://mathworld.wolfram.com/vanderWaerdensTheorem.html), using the given properties, but I haven't any ideas.

Title: Re: Angles in GP
Post by ronnodas on Mar 9^th, 2009, 9:58am

Yeah, the problems are similar. Both of these can be solved with [hide]7 points in very symmetric configurations[/hide].