Difference between revisions of "2006 IMO Problems"
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Let <math>ABC</math> be a triangle with incentre <math>I.</math> A point <math>P</math> in the interior of the triangle satisfies <math>\angle PBA + \angle PCA = \angle PBC + \angle PCB</math>. | Let <math>ABC</math> be a triangle with incentre <math>I.</math> A point <math>P</math> in the interior of the triangle satisfies <math>\angle PBA + \angle PCA = \angle PBC + \angle PCB</math>. | ||
Show that <math>AP \ge AI,</math> and that equality holds if and only if <math>P = I.</math> | Show that <math>AP \ge AI,</math> and that equality holds if and only if <math>P = I.</math> | ||
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+ | [[2006 IMO Problems/Problem 1|Solution]] | ||
==Problem 2== | ==Problem 2== |
Latest revision as of 04:17, 20 February 2025
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 |