IMO 1981 (XXII International Mathematical Olympiad)

Foot of perpendiculars from point to triangle sides ratio

$P$ is a point inside a given triangle $ABC$, $D$, $E$, $F$ are the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$ respectively. Find all $P$ for which $\frac{BC}{PD}+\frac{CA}{PE}+\frac{AB}{PF}$ is least.

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

Foot of perpendiculars from point to triangle sides ratio

$P$ is a point inside a given triangle $ABC$, $D$, $E$, $F$ are the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$ respectively. Find all $P$ for which $\frac{BC}{PD}+\frac{CA}{PE}+\frac{AB}{PF}$ is least.

src_imho_1981__Q01

Smallest element subset mean function identity

Let $1\le r\le n$ and consider all subsets of $r$ elements of the set $\{1,2,\dots ,n\}$. Each of these subsets has a smallest member. Let $F(n,r)$ denote the arithmetic mean of these smallest numbers. Prove that $F(n,r)=\frac{n+1}{r+1}.$

Topic: Combinatoria, Teoria dei Numeri Metodo: Doppio conteggio, Induzione, Telescoping Abilita: Conteggio sistematico, Manipolazione algebrica, Riconoscimento di pattern Area: Combinatoria, Logica e Probabilita, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Smallest element subset mean function identity

Let $1\le r\le n$ and consider the subsets of $r$ elements of the set $\{1,2,\dots ,n\}$. Each of these subsets has a smallest member. Let $F(n,r)$ denotes the arithmetic mean of these smallest numbers. Prove that $F(n,r)=\frac{n+1}{r+1}.$

src_imho_1981__Q02

Maximum value of m^n + n^m with integer constraint

Determine the maximum value of $m^n+n^m$, where $m$ and $n$ are integers satisfying $m,n\in \{1,2,\dots ,1981\}$ and $(n^2-mn-m^2{)}^2=1$.

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

Maximum value of m^n + n^m with integer constraint

Determine the maximum value of $m^n+n^m$, where $m$ and $n$ are integers satisfying $m,n\in \{1,2,\dots ,1981\}$ and $(n^2-mn-m^2{)}^2=1$.

src_imho_1981__Q03

Largest number in set divides LCM of remaining n-1

(a) For which values of $n>2$ is there a set of $n$ consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining $n-1$ numbers?

(b) For which values of $n>2$ is there exactly one such set having the stated property?

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

Largest number in set divides LCM of remaining n-1

(a) For which values of $n>2$ is there a set of $n$ consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining $n-1$ numbers?

(b) For which values of $n>2$ is there exactly one such set having the stated property?

src_imho_1981__Q04

Three congruent circles inscribed in triangle, circumcenter collinear

Three congruent circles have a common point $O$ and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle and the point $O$ are collinear.

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

*Three congruent circles inscribed in a triangle, circumcenter hill *

Three congruent circles have a common point $O$ and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle and the point $O$ are hilly.

src_imho_1981__Q05

Functional equation determine f(4,1981)

The function $f(x,y)$ satisfies (1) $f(0,y)=y+1$, (2) $f(x+1,0)=f(x,1)$, (3) $f(x+1,y+1)=f(x,f(x+1,y))$, for all non-negative integers $x$, $y$. Determine $f(4,1981)$.

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

*Functional equation to determine f(4,1981) *

The function $f(x,y)$ satisfies (1) $f(0,y)=y+1$, (2) $f(x+1,0)=f(x,1)$, (3) $f(x+1,y+1)=f(x,f(x+1,y))$, for all non-negative integers $x$, $y$. Determine $f(4,1981)$.

src_imho_1981__Q06