IMO 1988 (XXIX International Mathematical Olympiad)

Coplanar circles, perpendicular, locus of midpoint

Consider two coplanar circles of radii $R$ and $r$ ($R>r$) with the same center. Let $P$ be a fixed point on the smaller circle and $B$ a variable point on the larger circle. The line $BP$ meets the larger circle again at $C$. The perpendicular $l$ to $BP$ at $P$ meets the smaller circle again at $A$. (If $l$ is tangent to the circle at $P$ then $A=P$.)

(i) Find the set of values of $B{C}^2+C{A}^2+A{B}^2$.

(ii) Find the locus of the midpoint of $BC$.

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

Coplanar circles, perpendicular, locus of midpoint

Consider two coplanar circles of radii $R$ and $r$ ($R>r$) with the same center. Let $P$ be a fixed point on the smaller circle and $B$ a variable point on the larger circle. The line $BP$ meets the larger circle again at $C$. The perpendicular $l$ to $BP$ at $P$ meets the smaller circle again at $A$. (If $l$ is tangent to the circle at $P$ then $A=P$.)

(i) Find the set of values of $B{C}^2+C{A}^2+A{B}^2$.

(ii) Find the locus of the midpoint of $BC$.

src_imho_1988__Q01

Subset family with intersection and element covering conditions

Let $n$ be a positive integer and let $A_1,A_2,\dots ,A_{2n+1}$ be subsets of a set $B$. Suppose that

(a) Each $A_i$ has exactly $2n$ elements,

(b) Each $A_i\cap A_j$ ($1\le i<j\le 2n+1$) contains exactly one element, and

(c) Every element of $B$ belongs to at least two of the $A_i$.

For which values of $n$ can one assign to every element of $B$ one of the numbers $0$ and $1$ in such a way that $A_i$ has $0$ assigned to exactly $n$ of its elements?

Topic: Combinatoria, Insiemi e funzioni Metodo: Casework, Doppio conteggio, Induzione Abilita: Conteggio sistematico, Ragionamento geometrico, Astrazione, Lettura attenta Area: Combinatoria, Logica e Probabilita, Algebra e Analisi Fonte: apri PDF p.1

Subset family with intersection and element covering conditions

Let $n$ be a positive integer and let $A_1,A_2,\dots ,A_{2n+1}$ be subsets of a set $B$. Suppose that

(a) Each $A_i$ has exactly $2n$ elements,

(b) Each $A_i\cap A_j$ ($1\le i<j\le 2n+1$) contains exactly one element and

(c) Every element of $B$ belongs to at least two of the $A_i$.

For which values of $n$ can one assign to every element of $B$ one of the numbers $0$ and $1$ in such a way that $A_i$ has $0$ assigned to exactly $n$ of its elements?

src_imho_1988__Q02

Functional equations on positive integers, count fixed points

A function $f$ is defined on the positive integers by $f(1)=1,f(3)=3,$ $f(2n)=f(n),$ $f(4n+1)=2f(2n+1)-f(n),$ $f(4n+3)=3f(2n+1)-2f(n),$ for all positive integers $n$.

Determine the number of positive integers $n$, less than or equal to $1988$, for which $f(n)=n$.

Topic: Teoria dei Numeri, Equazioni funzionali Metodo: Induzione, Ricorsione, Casework Abilita: Riconoscimento di pattern, Manipolazione algebrica, Conteggio sistematico, Lettura attenta Area: Aritmetica e Teoria dei Numeri, Algebra e Analisi Fonte: apri PDF p.1

Functional equations on positive integers, count fixed points

A function $f$ is defined on the positive integers by $f(1)=1,f(3)=3,$ $f(2n)=f(n),$ $f(4n+1)=2f(2n+1)-f(n),$ $f(4n+3)=3f(2n+1)-2f(n),$ for all positive integers $n$.

Determine the number of positive integers $n$, less than or equal to $1988$, for which $f(n)=n$.

src_imho_1988__Q03

Sum inequality yields union of intervals of total length 1988

Show that the set of real numbers $x$ which satisfy the inequality ${\sum }_{k=1}^{70}\frac{k}{x-k}\ge \frac{5}{4}$ is a union of disjoint intervals, the sum of whose lengths is $1988$.

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

Sum inequality yields union of intervals of total length 1988

Show that the set of real numbers $x$ which satisfy the inequality ${\sum }_{k=1}^{70}\frac{k}{x-k}\ge \frac{5}{4}$ is a union of disjoint intervals, the sum of whose lengths is $1988$.

src_imho_1988__Q04

Right triangle, incenters line meets sides, area inequality

$ABC$ is a triangle right-angled at $A$, and $D$ is the foot of the altitude from $A$. The straight line joining the incenters of the triangles $ABD$, $ACD$ intersects the sides $AB$, $AC$ at the points $K$, $L$ respectively. $S$ and $T$ denote the areas of the triangles $ABC$ and $AKL$ respectively. Show that $S\ge 2T$.

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

Right triangle, incenters line meets sides, area of inequality

$ABC$ is a right-angled triangle at $A$, and $D$ is the foot of the altitude from $A$. The straight line joining the incenters of the triangles $ABD$, $ACD$ intersects the sides $AB$, $AC$ at the points $K$, $L$ respectively. $S$ and $T$ denote the areas of the triangles $ABC$ and $AKL$ respectively. Show that$S\ge 2T$.

src_imho_1988__Q05

ab+1 divides a²+b², quotient is a perfect square

Let $a$ and $b$ be positive integers such that $ab+1$ divides $a^2+b^2$. Show that $\frac{a^2+b^2}{ab+1}$ is the square of an integer.

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

ab+1 divides a2+b2, the quotient is a perfect square

Let $a$ and $b$ be positive integers such that $ab+1$ divides $a^2+b^2$. Show that $\frac{a^2+b^2}{ab+1}$ is the square of an integer.

src_imho_1988__Q06