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PrintIranian Mathematical Olympiad
Iran geometry
Problem
Morteza placed points on the plane. Then he writes down all the areas of the triangles constructed by these points (a total of numbers). Prove that he can always assign positive or negative signs to these numbers such that the sum of these numbers becomes zero.
Solution
First of all consider points , , and .
i) If convex hull of these points is a triangle (without loss of generality let be the inner point)
ii) If convex hull of these points is a quadrilateral
Let , , , , be problem's points. Now consider these sets of points, it is easy to see that each triangle of triangle is exactly in one of these sets, and from above it is possible to put and between areas in every set such that the result becomes zero, hence we are done.
i) If convex hull of these points is a triangle (without loss of generality let be the inner point)
ii) If convex hull of these points is a quadrilateral
Let , , , , be problem's points. Now consider these sets of points, it is easy to see that each triangle of triangle is exactly in one of these sets, and from above it is possible to put and between areas in every set such that the result becomes zero, hence we are done.
Techniques
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