Difference between revisions of "1983 IMO Problems/Problem 4"
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==Problem== | ==Problem== | ||
Let <math>ABC</math> be an equilateral triangle and <math>\mathcal{E}</math> the set of all points contained in the three segments <math>AB</math>, <math>BC</math> and <math>CA</math> (including <math>A</math>, <math>B</math> and <math>C</math>). Determine whether, for every partition of <math>\mathcal{E}</math> into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle. Justify your answer. | Let <math>ABC</math> be an equilateral triangle and <math>\mathcal{E}</math> the set of all points contained in the three segments <math>AB</math>, <math>BC</math> and <math>CA</math> (including <math>A</math>, <math>B</math> and <math>C</math>). Determine whether, for every partition of <math>\mathcal{E}</math> into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle. Justify your answer. | ||
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==Solution== | ==Solution== |
Revision as of 16:20, 22 August 2017
Problem
Let be an equilateral triangle and
the set of all points contained in the three segments
,
and
(including
,
and
). Determine whether, for every partition of
into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle. Justify your answer.