This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 10

2006 IMO Shortlist, 6
Tags: inequalities , function , algebra , IMO Shortlist , IMO , IMO 2006 , Hi
Determine the least real number $M$ such that the inequality \[|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \leq M(a^{2}+b^{2}+c^{2})^{2}\] holds for all real numbers $a$, $b$ and $c$.

2006 IMO, 2
Tags: combinatorics , polygon , Extremal combinatorics , IMO , IMO 2006 , IMO Shortlist , Dusan Djukic
Let $P$ be a regular $2006$-gon. A diagonal is called [i]good[/i] 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 [i]good[/i]. 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.

2006 IMO, 1
Tags: geometry , incenter , circumcircle , IMO Shortlist , IMO 2006 , IMO
Let $ABC$ be triangle with incenter $I$. A point $P$ in the interior of the triangle satisfies \[\angle PBA+\angle PCA = \angle PBC+\angle PCB.\] Show that $AP \geq AI$, and that equality holds if and only if $P=I$.

2006 IMO Shortlist, 1
Tags: geometry , incenter , circumcircle , IMO Shortlist , IMO 2006 , IMO
Let $ABC$ be triangle with incenter $I$. A point $P$ in the interior of the triangle satisfies \[\angle PBA+\angle PCA = \angle PBC+\angle PCB.\] Show that $AP \geq AI$, and that equality holds if and only if $P=I$.

2006 IMO, 5
Tags: algebra , polynomial , roots , IMO , IMO 2006 , IMO Shortlist , Dan Schwarz
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 $Q(t) = t$.

2006 IMO Shortlist, 10
Tags: geometry , area , geometric inequality , IMO , IMO 2006 , IMO Shortlist , Dusan Djukic
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 $P$.

2006 IMO Shortlist, 2
Tags: combinatorics , polygon , Extremal combinatorics , IMO , IMO 2006 , IMO Shortlist , Dusan Djukic
Let $P$ be a regular $2006$-gon. A diagonal is called [i]good[/i] 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 [i]good[/i]. 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.

2006 IMO Shortlist, 4
Tags: algebra , polynomial , roots , IMO , IMO 2006 , IMO Shortlist , Dan Schwarz
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 $Q(t) = t$.

2006 IMO, 6
Tags: geometry , area , geometric inequality , IMO , IMO 2006 , IMO Shortlist , Dusan Djukic
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 $P$.

2006 IMO, 3
Tags: inequalities , function , algebra , IMO Shortlist , IMO , IMO 2006 , Hi
Determine the least real number $M$ such that the inequality \[|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \leq M(a^{2}+b^{2}+c^{2})^{2}\] holds for all real numbers $a$, $b$ and $c$.