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Difference between revisions of "2006 IMO Problems"

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Latest revision as of 04:17, 20 February 2025

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 $P = I.$

Solution

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 $P.$

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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