IMO 1970 (XII International Mathematical Olympiad)

Ratio of inradii equals ratio of exradii in triangle

Let $M$ be a point on the side $AB$ of $△ABC$. Let $r_1$, $r_2$, $r$ be the radii of the inscribed circles of triangles $AMC$, $BMC$ and $ABC$. Let $q_1$, $q_2$, $q$ be the radii of the escribed circles of the same triangles that lie in the angle $ACB$. Prove that $\frac{r_1}{q_1}\cdot \frac{r_2}{q_2}=\frac{r}{q}.$

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

Ratio of inradii equals ratio of exradii in triangle

Let $M$ be a point on the side $AB$ of $△ABC$. Let $r_1$, $r_2$, $r$ be the radii of the inscribed circles of triangles $AMC$, $BMC$ and $ABC$. Let $q_1$, $q_2$, $q$ be the radii of the inscribed circles of the same triangles that lie in the angle $ACB$. Prove that $\frac{r_1}{q_1}\cdot \frac{r_2}{q_2}=\frac{r}{q}.$

src_imho_1970__Q01

Representation of numbers in two bases; compare sizes via digits

Let $a$, $b$ be integers greater than 1, and let $a$ and $b$ be the bases of two number systems. $A_{n-1}$ and $A_n$ are numbers in the system with base $a$; $B_{n-1}$ and $B_n$ are the numbers in the system with base $b$; these are related as follows: $A_n=x_n{x}_{n-1}\cdots x_1{x}_0,A_{n-1}=x_{n-1}{x}_{n-2}\cdots x_1{x}_0,$ $B_n=x_n{x}_{n-1}\cdots x_1{x}_0,B_{n-1}=x_{n-1}{x}_{n-2}\cdots x_1{x}_0,$ $x_n\ne 0,x_{n-1}\ne 0.$ Prove that $\frac{A_{n-1}}{A_n}<\frac{B_{n-1}}{B_n}\text{ if and only if }a>b.$

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

Representation of numbers in two bases; compare sizes by digits

Let $a$, $b$ be integers greater than 1, and let $a$ and $b$ be the bases of two number systems. $A_{n-1}$ and $A_n$ are numbers in the system with base $a$; $B_{n-1}$ and $B_n$ are the numbers in the system with base $b$; these are related as follows: $A_n=x_n{x}_{n-1}\cdots x_1{x}_0,A_{n-1}=x_{n-1}{x}_{n-2}\cdots x_1{x}_0,$ $B_n=x_n{x}_{n-1}\cdots x_1{x}_0,B_{n-1}=x_{n-1}{x}_{n-2}\cdots x_1{x}_0,$ $x_n\ne 0,x_{n-1}\ne 0.$ Prove that $\frac{A_{n-1}}{A_n}<\frac{B_{n-1}}{B_n}\text{ if and only if }a>b.$

src_imho_1970__Q02

Sequence satisfying sum condition; bounds and limit of derived sequence

The real numbers $a_0,a_1,a_2,\dots$ satisfy the condition: $1=a_0\le a_1\le a_2\le \cdots .$ The numbers $b_1,b_2,b_3,\dots$ are defined by $b_n={\sum }_{k=1}^n\left(1-\frac{a_{k-1}}{a_k}\right)\frac{1}{\sqrt{a_k}}.$ (a) Prove that $0\le b_n<2$ for all $n$.

(b) Given $c$ with $0\le c<2$, prove that there exist numbers $a_0,a_1,a_2,\dots$ with the above properties such that $b_n>c$ for large enough $n$.

Topic: Algebra, Disuguaglianze Metodo: Disuguaglianze, Induzione, Telescoping Abilita: Manipolazione algebrica, Stima, Lettura attenta Area: Algebra e Analisi Fonte: apri PDF p.1

Sequence satisfying sum condition; bounds and limit of derived sequence

The real numbers $a_0,a_1,a_2,\dots$ satisfy the condition: $1=a_0\le a_1\le a_2\le \cdots .$ The numbers $b_1,b_2,b_3,\dots$ are defined by $b_n={\sum }_{k=1}^n\left(1-\frac{a_{k-1}}{a_k}\right)\frac{1}{\sqrt{a_k}}.$ (a) Prove that $0\le b_n<2$ for all $n$.

(b) Given $c$ with $0\le c<2$, prove that there exist numbers $a_0,a_1,a_2,\dots$ with the above properties such that $b_n>c$ for large enough $n$.

src_imho_1970__Q03

Partition of positive integers into two sets with equal products

Find the set of all positive integers $n$ with the property that the set $\{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two sets such that the product of the numbers in one set equals the product of the numbers in the other set.

Topic: Teoria dei Numeri, Combinatoria Metodo: Casework, Fattorizzazione Abilita: Lettura attenta, Manipolazione algebrica, Riconoscimento di pattern, Conteggio sistematico Area: Aritmetica e Teoria dei Numeri, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Partition of positive integers into two sets with equal products

Find the set of all positive integers $n$ with the property that the set $\{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two sets such that the product of the numbers in one set equals the product of the numbers in the other set.

src_imho_1970__Q04

Tetrahedron with right angle BDC and orthocentric foot; prove inequality

In the tetrahedron $ABCD$, angle $BDC$ is a right angle. Suppose that the foot $H$ of the perpendicular from $D$ to the plane $ABC$ is the intersection of the altitudes of $△ABC$. Prove that $(AB+BC+CA{)}^2\le 6(A{D}^2+B{D}^2+C{D}^2).$ For what tetrahedra does equality hold?

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

Tetrahedron with right angle BDC and orthocentric foot; evidence of inequality

In the tetrahedron $ABCD$, angle $BDC$ is a right angle. Suppose that the foot $H$ of the perpendicular from $D$ to the plane $ABC$ is the intersection of the altitudes of $△ABC$. Prove that for what tetrahedra does equality hold?

src_imho_1970__Q05

100 points in general position; at most 70% of triangles are acute

In a plane there are 100 points, no three of which are collinear. Consider all possible triangles having these points as vertices. Prove that no more than $70\%$ of these triangles are acute-angled.

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

100 points in general position; at most 70% of triangles are sharp

In a plane there are 100 points, no three of which are collinear. Consider all possible triangles having these points as vertices. Prove that no more than $70\%$ of these triangles are acute-angled.

src_imho_1970__Q06