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 sided polygon. 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
Determine the least real number such that the inequality holds for all real numbers and
Problem 4
Determine all pairs of integers such that
Problem 5
Let be a polynomial of degree with integer coefficients, and let be a positive integer. Consider the polynomial , where occurs times. Prove that there are at most integers such that .
Problem 6
Let be a convex -sided polygon with vertices and sides For a given side let be the maximum possible area of a triangle with vertices among and with as a side. Show that the sum of the areas is at least twice the area of
See Also
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 |