2006 IMO Problems

Problem 1

Let $ABC$ 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 $P$ 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.

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2006 IMO Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

See Also