Difference between revisions of "2006 IMO Problems"
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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>\ | + | 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 | + | 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 14:55, 23 July 2019
Problem 1
Let 
be a triangle with incentre 
A point 
in the interior of the triangle satisfies 
.
Show that 
and that equality holds if and only if 
Problem 2
Let be a regular 2006-gon. A diagonal of is called good if its endpoints divide the boundary of into two parts, each composed of an odd number of sides of . The sides of are also called good. Suppose has been dissected into triangles by 2003 diagonals, no two of which have a common point in the interior of . Find the maximum number of isosceles triangles having two good sides that could appear in such a configuration.
