IMO 1991 (XXXII International Mathematical Olympiad)

Incenter ratio inequality with angle bisectors

Given a triangle $ABC$, let $I$ be the center of its inscribed circle. The internal bisectors of the angles $A$, $B$, $C$ meet the opposite sides in $A^{\prime }$, $B^{\prime }$, $C^{\prime }$ respectively. Prove that $\frac{1}{4}<\frac{AI\cdot BI\cdot CI}{A{A}^{\prime }\cdot B{B}^{\prime }\cdot C{C}^{\prime }}\le \frac{8}{27}.$

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

Incenter ratio inequality with angle bisectors

Given a triangle $ABC$, let $I$ be the center of its inscribed circle. The internal bisectors of the $A$, $B$, $C$ meet the opposite sides in $A^{\prime }$, $B^{\prime }$, $C^{\prime }$ respectively. Prove that $\frac{1}{4}<\frac{AI\cdot BI\cdot CI}{A{A}^{\prime }\cdot B{B}^{\prime }\cdot C{C}^{\prime }}\le \frac{8}{27}.$

src_imho_1991__Q01

Arithmetic residues mod n implies n is 1, 2, 4, prime power, or double

Let $n>0$ be an integer and let $a_1,a_2,\dots ,a_k$ be all the natural numbers less than and relatively prime to $n$. If $a_2-a_1=a_3-a_2=\cdots =a_k-a_{k-1}>0,$ prove that $n$ must be either $1$, $2$, $4$, a power of an odd prime, or twice a power of an odd prime.

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

Arithmetic residues mod n implies n is 1, 2, 4, prime power, or double

Let $n>0$ be an integer and let $a_1,a_2,\dots ,a_k$ be all the natural numbers less than and relatively prime to $n$. If $a_2-a_1=a_3-a_2=\cdots =a_k-a_{k-1}>0,$ proves that $n$ must be either $1$, $2$, $4$, a power of an odd prime, or twice a power of an odd prime.

src_imho_1991__Q02

Smallest n-element subset of {1,…,280} with five pairwise coprime members

Let $S=\{1,2,3,\dots ,280\}$. Find the smallest integer $n$ such that each $n$-element subset of $S$ contains five numbers which are pairwise relatively prime.

Topic: Combinatoria, Teoria dei Numeri Metodo: Estremalità, Principio dei cassetti, Conteggio Abilita: Conteggio sistematico, Riconoscimento di pattern, Casework accurato Area: Combinatoria, Logica e Probabilita, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Smallest n-element subset of {1,…,280} with five pairwise coprime members

Let $S=\{1,2,3,\dots ,280\}$. Find the smallest integer $n$ such that each $n$-element subset of $S$ contains five numbers which are pairwise relatively prime.

src_imho_1991__Q03

Label edges of connected graph so gcd at each vertex equals 1

Suppose $G$ is a connected graph with $k$ edges. Prove that it is possible to label the edges $1,2,\dots ,k$ in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to $1$.

[A graph consists of a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of distinct vertices $u$, $v$ belongs to at most one edge. The graph $G$ is connected if for each pair of distinct vertices $x$, $y$ there is some sequence of vertices $x=v_0,v_1,v_2,\dots ,v_m=y$ such that each pair $v_i,v_{i+1}$ $(0\le i<m)$ is joined by an edge of $G$.]

Topic: Combinatoria Metodo: Grafi, Induzione Abilita: Astrazione, Modellizzazione, Ragionamento geometrico Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Label edges of connected graph so gcd at each vertex equals 1

Suppose $G$ is a connected graph with $k$ edges. Prove that it is possible to label the edges $1,2,\dots ,k$ in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to $1$.

[A graph consists of a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of distinct vertices $u$, $v$ belongs to at most one edge. The graph $G$ is connected if for each pair of distinct vertices $x$, $y$ there is some sequence of vertices $x=v_0,v_1,v_2,\dots ,v_m=y$ such that each pair $v_i,v_{i+1}$ $(0\le i<m)$ is joined by an edge of $G$.]

src_imho_1991__Q04

Interior point of triangle has one angle among PAB, PBC, PCA at most 30°

Let $ABC$ be a triangle and $P$ an interior point of $ABC$. Show that at least one of the angles $\angle PAB$, $\angle PBC$, $\angle PCA$ is less than or equal to $30^{\circ }$.

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

Interior point of triangle has one angle among PAB, PBC, PCA at most 30°

Let $ABC$ be a triangle and $P$ an interior point of $ABC$. Show that at least one of the angles $\angle PAB$, $\angle PBC$, $\angle PCA$ is less than or equal to $30^{\circ }$.

src_imho_1991__Q05

Construct bounded sequence with |x_i - x_j||i-j|^a ≥ 1

An infinite sequence $x_0,x_1,x_2,\dots$ of real numbers is said to be bounded if there is a constant $C$ such that $|x_i|\le C$ for every $i\ge 0$.

Given any real number $a>1$, construct a bounded infinite sequence $x_0,x_1,x_2,\dots$ such that $|x_i-x_j|\cdot |i-j{|}^a\ge 1$ for every pair of distinct nonnegative integers $i$, $j$.

Topic: Algebra, Insiemi e funzioni Metodo: Ricorsione, Induzione Abilita: Astrazione, Modellizzazione, Riconoscimento di pattern, Manipolazione algebrica Area: Algebra e Analisi Fonte: apri PDF p.1

*Construct bounded sequence with — x_j>

An infinite sequence $x_0,x_1,x_2,\dots$ of real numbers is said to be bounded if there is a constant $C$ such that $|x_i|\le C$ for every $i\ge 0$.

Given any real number $a>1$, construct a bounded infinite sequence $x_0,x_1,x_2,\dots$ such that $|x_i-x_j|\cdot |i-j{|}^a\ge 1$ for every pair of distinct nonnegative integers $i$, $j$.

src_imho_1991__Q06