IMO 1973 (XV International Mathematical Olympiad)

Unit vectors from O: sum of lengths condition with parity

Point $O$ lies on line $g$. $\overrightarrow{O{P}_1}$, $\overrightarrow{O{P}_2}$, $\dots$, $\overrightarrow{O{P}_n}$ are unit vectors such that points $P_1,P_2,\dots ,P_n$ all lie in a plane containing $g$. Prove that if $n$ is odd, $|\overrightarrow{O{P}_1}+\overrightarrow{O{P}_2}+\cdots +\overrightarrow{O{P}_n}|\ge 1.$ Here $|\overrightarrow{OM}|$ denotes the length of vector $\overrightarrow{OM}$.

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

Unit vectors from O: sum of lengths condition with parity

Point $O$ lies on line $g$. $\overrightarrow{O{P}_1}$, $\overrightarrow{O{P}_2}$, $\dots$, $\overrightarrow{O{P}_n}$ are unit vectors such that points $P_1,P_2,\dots ,P_n$ all lie in a plane containing $g$. Prove that if $n$ is odd, $|\overrightarrow{O{P}_1}+\overrightarrow{O{P}_2}+\cdots +\overrightarrow{O{P}_n}|\ge 1.$ Here $|\overrightarrow{OM}|$ denotes the length of vector $\overrightarrow{OM}$.

src_imho_1973__Q01

Finite set M with parallel midpoint lines property

Determine whether or not there exists a finite set $M$ of points in space not lying in the same plane such that, for any two points $A$ and $B$ of $M$, one can select two other points $C$ and $D$ of $M$ so that lines $AB$ and $CD$ are parallel and not coincident.

Topic: Geometria solida, Combinatoria Metodo: Casework, Estremalità Abilita: Ragionamento geometrico, Astrazione, Lettura attenta Area: Geometria, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Finite set M with parallel midpoint lines property

Determine whether or not there exists a finite set $M$ of points in space not lying in the same plane such that, for any two points $A$ and $B$ of $M$, one can select two other points $C$ and $D$ of $M$ so that lines $AB$ and $CD$ are parallel and not coincident.

src_imho_1973__Q02

Real a,b: quartic equation has at least one real root, find min value

Let $a$ and $b$ be real numbers for which the equation $x^4+a{x}^3+b{x}^2+ax+1=0$ has at least one real solution. For all such pairs $(a,b)$, find the minimum value of $b$.

Topic: Algebra, Disuguaglianze Metodo: Disuguaglianze, Estremalità, Fattorizzazione Abilita: Manipolazione algebrica, Riconoscimento di pattern, Stima Area: Algebra e Analisi Fonte: apri PDF p.1

Real a,b: quartic equation has at least one real root, find min value

Let $a$ and $b$ be real numbers for which the equation $x^4+a{x}^3+b{x}^2+ax+1=0$ has at least one real solution. For all such pairs $(a,b)$, find the minimum value of $b$.

src_imho_1973__Q03

Soldier finds path of least distance through equilateral triangle

A soldier needs to check on the presence of mines in a region having the shape of an equilateral triangle. The radius of action of his detector is equal to half the altitude of the triangle. The soldier leaves from one vertex of the triangle. What path should he follow in order to travel the least possible distance and still accomplish his mission?

Topic: Geometria piana Metodo: Estremalità, Simmetria Abilita: Ragionamento geometrico, Modellizzazione, Stima Area: Geometria Fonte: apri PDF p.1

Soldier finds path of least distance through equilateral triangle

A soldier needs to check on the presence of mines in a region having the shape of an equilateral triangle. The radius of action of his detector is equal to half the altitude of the triangle. The soldier leaves from one vertex of the triangle. What path should he follow in order to travel the least possible distance and still accomplish his mission?

src_imho_1973__Q04

Functions of the form f(x)=ax+b closed under composition and inverse

$G$ is a set of non-constant functions of the real variable $x$ of the form $f(x)=ax+b,a\text{ and }b\text{ are real numbers,}$ and $G$ has the following properties:

(a) If $f$ and $g$ are in $G$, then $g\circ f$ is in $G$; here $(g\circ f)(x)=g[f(x)]$.

(b) If $f$ is in $G$, then its inverse $f^{-1}$ is in $G$; here the inverse of $f(x)=ax+b$ is $f^{-1}(x)=(x-b)/a$.

(c) For every $f$ in $G$, there exists a real number $x_f$ such that $f(x_f)=x_f$.

Prove that there exists a real number $k$ such that $f(k)=k$ for all $f$ in $G$.

Topic: Insiemi e funzioni, Algebra Metodo: Invarianti, Simmetria Abilita: Astrazione, Manipolazione algebrica, Ragionamento geometrico Area: Algebra e Analisi Fonte: apri PDF p.1

Functions of the form f(x)=ax+b closed under composition and inverse

$G$ is a set of non-constant functions of the real variable $x$ of the form $f(x)=ax+b,a\text{ and }b\text{ are real numbers,}$ and $G$ has the following properties:

If $f$ and $g$ are in $G$, then $g\circ f$ is in $G$; here $(g\circ f)(x)=g[f(x)]$.

(b) If $f$ is in $G$, then its inverse $f^{-1}$ is in $G$; here the inverse of $f(x)=ax+b$ is $f^{-1}(x)=(x-b)/a$.

(c) For every $f$ in $G$, there exists a real number $x_f$ such that $f(x_f)=x_f$.

Prove that there exists a real number $k$ such that $f(k)=k$ for all $f$ in $G$.

src_imho_1973__Q05

Construct n numbers b_k satisfying bounds relative to given a_k and q

Let $a_1,a_2,\dots ,a_n$ be $n$ positive numbers, and let $q$ be a given real number such that $0<q<1$. Find $n$ numbers $b_1,b_2,\dots ,b_n$ for which

(a) $a_k<b_k$ for $k=1,2,\cdots ,n$,

(b) $q<\frac{b_{k+1}}{b_k}<\frac{1}{q}$ for $k=1,2,\dots ,n-1$,

(c) $b_1+b_2+\cdots +b_n<\frac{1+q}{1-q}(a_1+a_2+\cdots +a_n)$.

Topic: Algebra, Disuguaglianze Metodo: method_casework, Disuguaglianze, Induzione Abilita: Manipolazione algebrica, Stima, Conteggio sistematico Area: Algebra e Analisi Fonte: apri PDF p.1

Construct n numbers b_k satisfying bounds relative to given a_k and q

Let $a_1,a_2,\dots ,a_n$ be $n$ positive numbers, and let $q$ be a given real number such that $0<q<1$. Find $n$ numbers $b_1,b_2,\dots ,b_n$ for which

(a) $a_k<b_k$ for $k=1,2,\cdots ,n$,

(b) $q<\frac{b_{k+1}}{b_k}<\frac{1}{q}$ for $k=1,2,\dots ,n-1$,

(c) $b_1+b_2+\cdots +b_n<\frac{1+q}{1-q}(a_1+a_2+\cdots +a_n)$.

src_imho_1973__Q06