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 I. A point P in the interior of the triangle satisfies <math>\anglePBA</math> + <math>\anglePCA</math> = <math>\anglePBC</math> + <math>\anglePCB</math>. |
Show that AP ≥ AI, and that equality holds if and only if P = I. | Show that AP ≥ AI, and that equality holds if and only if P = I. | ||
Revision as of 23:08, 14 February 2019
Problem 1
Let 
be a triangle with incentre I. A point P in the interior of the triangle satisfies $\anglePBA$ (Error compiling LaTeX. Unknown error_msg) + $\anglePCA$ (Error compiling LaTeX. Unknown error_msg) = $\anglePBC$ (Error compiling LaTeX. Unknown error_msg) + $\anglePCB$ (Error compiling LaTeX. Unknown error_msg).
Show that AP ≥ AI, and that equality holds if and only if P = I.
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.
