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Print11th Junior Balkan Mathematical Olympiad
Greece counting and probability
Problem
Given are 50 points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least 130 scalene triangles with vertices of that color.
Solution
Since , according to the pigeonhole principle we will have at least 13 points colored in the same color. Using these 13 points we construct different triangles, since there are no three collinear points.
We will prove that there are at most isosceles triangles. In fact, there are different line segments. Each of them can be the basis at most two different isosceles triangles (because there are not three collinear points). Therefore there are at most isosceles triangles. Thus we can construct at least scalene triangles.
We will prove that there are at most isosceles triangles. In fact, there are different line segments. Each of them can be the basis at most two different isosceles triangles (because there are not three collinear points). Therefore there are at most isosceles triangles. Thus we can construct at least scalene triangles.
Techniques
Pigeonhole principleColoring schemes, extremal argumentsTriangles