Difference between revisions of "2006 IMO Problems"

Latest revision as of 22:18, 12 April 2021

Problem 1

Let $ABC$ be a triangle with incentre $I.$ A point $P$ in the interior of the triangle satisfies $\angle PBA + \angle PCA = \angle PBC + \angle PCB$ . Show that $AP \ge AI,$ and that equality holds if and only if $P = I.$

Problem 2

Let $P$ be a regular 2006 sided polygon. 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

Determine the least real number $M$ such that the inequality $\left| ab\left(a^{2}-b^{2}\right)+bc\left(b^{2}-c^{2}\right)+ca\left(c^{2}-a^{2}\right)\right|\leq M\left(a^{2}+b^{2}+c^{2}\right)^{2}$ holds for all real numbers $a,b$ and $c$

Problem 4

Determine all pairs $(x, y)$ of integers such that $1+2^{x}+2^{2x+1}= y^{2}.$

Problem 5

Let $P(x)$ be a polynomial of degree $n>1$ with integer coefficients, and let $k$ be a positive integer. Consider the polynomial $Q(x) = P( P ( \ldots P(P(x)) \ldots ))$ , where $P$ occurs $k$ times. Prove that there are at most $n$ integers $t$ such that $Q(t)=t$ .

Problem 6

Let $P$ be a convex $n$ -sided polygon with vertices $V_1, V_2, \dots, V_n,$ and sides $S_1, S_2, \dots, S_n.$ For a given side $S_i,$ let $A_i$ be the maximum possible area of a triangle with vertices among $V_1, V_2, \dots, V_n$ and with $S_i$ as a side. Show that the sum of the areas $A_1, A_2, \dots, A_n$ is at least twice the area of $P.$

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 ==Problem 2==
-Let <math>P</math> be a regular 2006-gon. 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.
+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==
-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.
+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 (Problems) • Resources
Preceded by 2005 IMO Problems	1 • 2 • 3 • 4 • 5 • 6	Followed by 2007 IMO Problems
All IMO Problems and Solutions

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