Difference between revisions of "1959–1966 IMO Longlist Problems/Czechoslovakia 1"
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== Resources == | == Resources == | ||
| + | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=16193 Discussion on AoPS/MathLinks] | ||
* [[1959–1966 IMO Longlist Problems]] | * [[1959–1966 IMO Longlist Problems]] | ||
[[Category:Olympiad Combinatorics Problems]] | [[Category:Olympiad Combinatorics Problems]] | ||
Latest revision as of 22:19, 28 July 2006
Problem
Given 
points in the plane such that no three of the points are collinear. Does there exist a circle passing through (at least) 3 of the given points and not containing any other of the 
points in its interior?
Solution
The answer is yes.
Since any finite set of at least three coplanar points is contained by a convex hull with vertices in the set of points, we can select adjacent points 
and 
on this convex hull. Clearly all of the other 
points will lie on the same side of the line 
. Of these other points, we select the point 
such that the angle 
is maximized. Then 
satisfy the conditions of the problem, because if there were some point 
inside the circle, since it would be on the same side of line as , the angle 
would be greater than the angle , which is a contradiction. Q.E.D.
