2006 IMO Problems
Problem 1
Let be a triangle with incentre
A point
in the interior of the triangle satisfies
.
Show that
and that equality holds if and only if
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.
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.