IMO 1989 (XXX International Mathematical Olympiad)

Partition {1,…,1989} into 117 equal-sum 17-element subsets

Prove that the set $\{1,2,\dots ,1989\}$ can be expressed as the disjoint union of subsets $A_i$ ($i=1,2,\dots ,117$) such that:

(i) Each $A_i$ contains 17 elements;

(ii) The sum of all the elements in each $A_i$ is the same.

Topic: Combinatoria, Teoria dei Numeri Metodo: method_casework, Invarianti Abilita: Modellizzazione, Conteggio sistematico, Manipolazione algebrica Area: Combinatoria, Logica e Probabilita, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Partition {1,…,1989} into 117 equal-sum 17-element subsets*

Prove that the set $\{1,2,\dots ,1989\}$ can be expressed as the disjoint union of subsets $A_i$ ($i=1,2,\dots ,117$) such that:

(i) Each $A_i$ contains 17 elements;

(ii) The sum of all the elements in each $A_i$ is the same.

src_imho_1989__Q01

Triangle bisectors, circumcircle points, area relations

In an acute-angled triangle $ABC$ the internal bisector of angle $A$ meets the circumcircle of the triangle again at $A_1$. Points $B_1$ and $C_1$ are defined similarly. Let $A_0$ be the point of intersection of the line $A{A}_1$ with the external bisectors of angles $B$ and $C$. Points $B_0$ and $C_0$ are defined similarly. Prove that:

(i) The area of the triangle $A_0{B}_0{C}_0$ is twice the area of the hexagon $A{C}_{1}B{A}_{1}C{B}_1$.

(ii) The area of the triangle $A_0{B}_0{C}_0$ is at least four times the area of the triangle $ABC$.

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

Triangle bisectors, circumcircle points, area relations

In an acute-angled triangle $ABC$ the internal bisector of angle $A$ meets the circumcircle of the triangle again at $A_1$. Points $B_1$ and $C_1$ are defined similarly. Let $A_0$ be the point of intersection of the line $A{A}_1$ with the external bisectors of angles $B$ and $C$. Points $B_0$ and $C_0$ are defined similarly. Prove that:

(i) The area of the triangle $A_0{B}_0{C}_0$ is twice the area of the hexagon $A{C}_{1}B{A}_{1}C{B}_1$.

(ii) The area of the triangle $A_0{B}_0{C}_0$ is at least four times the area of the triangle $ABC$.

src_imho_1989__Q02

Point set with equidistance condition implies k < 1/2 + sqrt(2n)

Let $n$ and $k$ be positive integers and let $S$ be a set of $n$ points in the plane such that

(i) No three points of $S$ are collinear, and

(ii) For any point $P$ of $S$ there are at least $k$ points of $S$ equidistant from $P$.

Prove that: $k<\frac{1}{2}+\sqrt{2n}.$

Topic: Combinatoria, Geometria piana Metodo: Doppio conteggio, Estremalità, Disuguaglianze Abilita: Modellizzazione, Stima, Ragionamento geometrico, Manipolazione algebrica Area: Combinatoria, Logica e Probabilita, Geometria Fonte: apri PDF p.1

*Point set with equidistance condition implies k < 1/2 + sqrt(2n) *

Let $n$ and $k$ be positive integers and let $S$ be a set of $n$ points in the plane such that

(i) No three points of $S$ are collinear, and

(ii) For any point $P$ of $S$ there are at least $k$ points of $S$ equidistant from $P$.

Prove that: $k<\frac{1}{2}+\sqrt{2n}.$

src_imho_1989__Q03

Convex quadrilateral with special point yields inequality 1/sqrt(h) >= 1/sqrt(AD)+1/sqrt(BC)

Let $ABCD$ be a convex quadrilateral such that the sides $AB$, $AD$, $BC$ satisfy $AB=AD+BC$. There exists a point $P$ inside the quadrilateral at a distance $h$ from the line $CD$ such that $AP=h+AD$ and $BP=h+BC$. Show that: $\frac{1}{\sqrt{h}}\ge \frac{1}{\sqrt{AD}}+\frac{1}{\sqrt{BC}}.$

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

*Convex quadrilateral with special point yields inequality 1/sqrt(h) >= 1/sqrt(AD) +1/sqrt(BC) *

Let $ABCD$ be a convex quadrilateral such that the sides $AB$, $AD$, $BC$ satisfy $AB=AD+BC$. There exists a point $P$ inside the quadrilateral at a distance $h$ from the line $CD$ such that $AP=h+AD$ and $BP=h+BC$. Show that:

src_imho_1989__Q04

For each n, find n consecutive integers none a prime power

Prove that for each positive integer $n$ there exist $n$ consecutive positive integers none of which is an integral power of a prime number.

Topic: Teoria dei Numeri, Combinatoria Metodo: method_casework, Congruenze, Fattorizzazione Abilita: Modellizzazione, Manipolazione algebrica, Riconoscimento di pattern Area: Aritmetica e Teoria dei Numeri, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

For each n, find n consecutive integers none at prime power

Prove that for every positive integer $n$ there exist $n$ consecutive positive integers none of which is an integral power of a prime number.

src_imho_1989__Q05

Permutations of {1,…,2n} with property P outnumber those without

A permutation $(x_1,x_2,\dots ,x_{2n})$ of the set $\{1,2,\dots ,2n\}$, where $n$ is a positive integer, is said to have property $P$ if $|x_i-x_{i+1}|=n$ for at least one $i$ in $\{1,2,\dots ,2n-1\}$. Show that, for each $n$, there are more permutations with property $P$ than without.

Topic: Combinatoria Metodo: Doppio conteggio, Biiezione, Inclusione-esclusione, Conteggio Abilita: Conteggio sistematico, Modellizzazione, Manipolazione algebrica, Ragionamento geometrico Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Permutations of {1,…,2n} with property P outnumber those without*

A permutation $(x_1,x_2,\dots ,x_{2n})$ of the set $\{1,2,\dots ,2n\}$, where $n$ is a positive integer, is said to have property $P$ if $|x_i-x_{i+1}|=n$ for at least one $i$ in $\{1,2,\dots ,2n-1\}$. Show that, for each $n$, there are more permutations with property $P$ than without.

src_imho_1989__Q06