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s65 - So you can suppose that no pair is colinear you next...

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Solution to Problem 65 No complete solutions were received, but partial solutions were sent in by Nathan Pauli, Ray Kremer, Mike Mitchell Correct solutions were received from Ivan Lisac, Burkart Venzke, Philippe Fondanaiche. A number of incorrect solutions were submitted. Here’s an interesting solution from Burkart Venzke. All nine points lie in the interior of the square, so pick any one and call it A . If any two of the other eight points are colinear with A, they form a traingle of area zero.
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Unformatted text preview: So you can suppose that no pair is colinear; you next pick any further point, call it B , and label the other points, C , D , E ,... clockwise and in order around A starting from B . Now look at the eight distinct and disjoint triangles with vertices ABC , ACD , ADE ,... The union of the eight triangles lies in the interior of the square, so the sum of the areas must be less than 1. One of the triangles must therefore have area less than 1/8. to this page....
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s65 - So you can suppose that no pair is colinear you next...

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