IMO 2006 (XLVII International Mathematical Olympiad)

Point in triangle interior with angle and equality condition

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 $P=I$.

Topic: Geometria piana Metodo: Disuguaglianze, Trigonometria Abilita: Ragionamento geometrico, Manipolazione algebrica, Lettura attenta Area: Geometria Fonte: apri PDF p.1

Point in triangle interior with angle and equality condition

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 $P=I$.

src_imho_2006__Q01

Regular 2006-gon diagonals: max isosceles triangles in dissection

Let $P$ be a regular 2006-gon. 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.

Topic: Combinatoria, Geometria piana Metodo: Casework, Estremalità, Conteggio Abilita: Conteggio sistematico, Riconoscimento di pattern, Ragionamento geometrico, Casework accurato Area: Combinatoria, Logica e Probabilita, Geometria Fonte: apri PDF p.1

Regular 2006-gon diagonals: max isosceles triangles in dissection

Let $P$ be a regular 2006-gon. 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.

src_imho_2006__Q02

Find least real M for an algebraic inequality in a, b, c

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

Topic: Disuguaglianze, Algebra Metodo: Disuguaglianze, Estremalità, Simmetria Abilita: Manipolazione algebrica, Stima, Ragionamento geometrico Area: Algebra e Analisi Fonte: apri PDF p.1

Find the least real M for an algebraic inequality in a, b, c

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

src_imho_2006__Q03

Find all integer pairs satisfying an exponential Diophantine equation

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

Topic: Teoria dei Numeri Metodo: Congruenze, Fattorizzazione, Casework Abilita: Manipolazione algebrica, Lettura attenta, Conteggio sistematico Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Find the integer pairs satisfying an exponential Diophantine equation

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

src_imho_2006__Q04

Iterated polynomial has at most n integer fixed points

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

Topic: Algebra, Teoria dei Numeri Metodo: Induzione, Fattorizzazione, Invarianti Abilita: Manipolazione algebrica, Astrazione, Ragionamento geometrico Area: Algebra e Analisi, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Iterated polynomial has at most n integer fixed points

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

src_imho_2006__Q05

Sum of max-triangle areas over sides is at least twice polygon area

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 $P$.

Topic: Geometria piana, Disuguaglianze Metodo: Estremalità, Disuguaglianze, Doppio conteggio Abilita: Ragionamento geometrico, Stima, Astrazione Area: Geometria, Algebra e Analisi Fonte: apri PDF p.1

Sum of max-triangle areas over sides is at least twice polygon area

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 $P$.

src_imho_2006__Q06