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Difference between revisions of "1983 IMO Problems/Problem 4"

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==Problem==
 
==Problem==
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Let <math>a</math>, <math>b</math> and <math>c</math> be positive integers, no two of which have a common divisor greater than <math>1</math>. Show that <math>2abc - ab - bc - ca</math> is the largest integer which cannot be expressed in the form <math>xbc + yca + zab</math>, where <math>x</math>, <math>y</math> and <math>z</math> are non-negative integers.
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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==
 
==Solution==

Revision as of 16:20, 22 August 2017

Problem

Let $ABC$ be an equilateral triangle and $\mathcal{E}$ the set of all points contained in the three segments $AB$, $BC$ and $CA$ (including $A$, $B$ and $C$). Determine whether, for every partition of $\mathcal{E}$ into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle. Justify your answer.

Solution

Solution

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