# Difference between revisions of "2006 IMO Problems"

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==Problem 1== | ==Problem 1== | ||

− | Let ABC be a triangle with incentre I. A point P in the interior of the triangle satisfies <PBA + <PCA = <PBC + <PCB. | + | Let <math>ABC be a triangle with incentre I. A point P in the interior of the triangle satisfies </math><PBA + <math><PCA = </math><PBC + $<PCB. |

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 23:51, 14 February 2019

## Problem 1

Let <PBA + <PBC + $<PCB. 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.