Difference between revisions of "2006 IMO Problems"

(Problem 1)
(Problem 1)
Line 1: Line 1:
 
==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><PBA</math> + <math><PCA</math> = <math><PBC</math> +  <math><PCB</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 00:08, 15 February 2019

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.

Problem 3

Problem 4

Problem 5

Problem 6

See Also