Difference between revisions of "2006 IMO Problems"

(Problem 1)
(Problem 1)
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==Problem 1==
 
==Problem 1==
Let <math>ABC</math> be a triangle with incentre I. A point P in the interior of the triangle satisfies <math>\anglePBA</math> + <math>\anglePCA</math> = <math>\anglePBC</math> + <math>\anglePCB</math>.
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Let <math>ABC</math> be a triangle with incentre <math>I.</math> A point <math>P</math> in the interior of the triangle satisfies <math>\angle PBA + \angle PCA = \angle PBC + \angle PCB</math>.
Show that AP AI, and that equality holds if and only if P = I.
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Show that <math>AP \ge AI,</math> and that equality holds if and only if <math>P = I.</math>
  
 
==Problem 2==
 
==Problem 2==

Revision as of 15:55, 23 July 2019

Problem 1

Let $ABC$ be a triangle with incentre $I.$ A point $P$ in the interior of the triangle satisfies $\angle PBA + \angle PCA = \angle PBC + \angle PCB$. Show that $AP \ge AI,$ and that equality holds if and only if $P = I.$

Problem 2

Let $P$ be a regular 2006-gon. A diagonal of $P$ is called good if its endpoints divide the boundary of $P$ into two parts, each composed of an odd number of sides of $P$. The sides of $P$ are also called good. Suppose $P$ has been dissected into triangles by 2003 diagonals, no two of which have a common point in the interior of $P$. Find the maximum number of isosceles triangles having two good sides that could appear in such a configuration.

Problem 3

Problem 4

Problem 5

Problem 6

See Also