Difference between revisions of "2006 IMO Problems"

Revision as of 08:24, 10 September 2020

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

Assign to each side b of a convex polygon P the maximum area of a triangle that has b as a side and is contained in P. Show that the sum of the areas assigned to the sides of P is at least twice the area of P.

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 * [[IMO Problems and Solutions]]
 * [[IMO]]
+{{IMO box|year=2006|before=[[2005 IMO Problems]]|after=[[2007 IMO Problems]]}}

2006 IMO (Problems) • Resources
Preceded by 2005 IMO Problems	1 • 2 • 3 • 4 • 5 • 6	Followed by 2007 IMO Problems
All IMO Problems and Solutions

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