2006 IMO Problems

Revision as of 23:52, 14 February 2019 by Nikbot05 (talk | contribs) (Problem 1)

Problem 1

Let $ABC$ be a triangle with incentre I. A point P in the interior of the triangle satisfies $<PBA$ + $<PCA$ = $<PBC$ + $<PCB$. Show that AP ≥ 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

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