Difference between revisions of "2006 IMO Problems"
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− | + | ==Problem 1== | |
+ | 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> | ||
+ | |||
+ | ==Problem 2== | ||
+ | Let <math>P</math> be a regular 2006 sided polygon. A diagonal of <math>P</math> is called good if its endpoints divide the boundary of <math>P</math> into two parts, each composed of an odd number of sides of <math>P</math>. The sides of <math>P</math> are also called good. Suppose <math>P</math> has been dissected into triangles by 2003 diagonals, no two of which have a common point in the interior of <math>P</math>. 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 <math>M</math> such that the inequality <cmath> \left| ab\left(a^{2}-b^{2}\right)+bc\left(b^{2}-c^{2}\right)+ca\left(c^{2}-a^{2}\right)\right|\leq M\left(a^{2}+b^{2}+c^{2}\right)^{2} </cmath> holds for all real numbers <math>a,b</math> and <math>c</math> | ||
+ | |||
+ | ==Problem 4== | ||
+ | Determine all pairs <math>(x, y)</math> of integers such that <cmath>1+2^{x}+2^{2x+1}= y^{2}.</cmath> | ||
+ | |||
+ | ==Problem 5== | ||
+ | Let <math>P(x)</math> be a polynomial of degree <math>n>1</math> with integer coefficients, and let <math>k</math> be a positive integer. Consider the polynomial <math>Q(x) = P( P ( \ldots P(P(x)) \ldots ))</math>, where <math>P</math> occurs <math>k</math> times. Prove that there are at most <math>n</math> integers <math>t</math> such that <math>Q(t)=t</math>. | ||
+ | |||
+ | ==Problem 6== | ||
+ | Let <math>P</math> be a convex <math>n</math>-sided polygon with vertices <math>V_1, V_2, \dots, V_n,</math> and sides <math>S_1, S_2, \dots, S_n.</math> For a given side <math>S_i,</math> let <math>A_i</math> be the maximum possible area of a triangle with vertices among <math>V_1, V_2, \dots, V_n</math> and with <math>S_i</math> as a side. Show that the sum of the areas <math>A_1, A_2, \dots, A_n</math> is at least twice the area of <math>P.</math> | ||
+ | |||
+ | ==See Also== | ||
+ | * [[2006 IMO]] | ||
+ | * [[IMO Problems and Solutions]] | ||
+ | * [[IMO]] | ||
+ | |||
+ | {{IMO box|year=2006|before=[[2005 IMO Problems]]|after=[[2007 IMO Problems]]}} |
Latest revision as of 23:18, 12 April 2021
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 |