IMO 1982 (XXIII International Mathematical Olympiad)

Functional equation for non-negative integers, find f(1982)

The function $f(n)$ is defined for all non-negative integers $n$ and takes on non-negative integer values. Also, for all $m,n$: $f(m+n)-f(m)-f(n)=0\text{ or }1$ $f(2)=0,f(3)>0,f(9999)=3333$.

Determine $f(1982)$.

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

*Functional equation for non-negative integers, found f(1982) *

The function $f(n)$ is defined for all non-negative integers $n$ and takes on non-negative integer values. Also, for all $m,n$: $f(m+n)-f(m)-f(n)=0\text{ or }1$ $f(2)=0,f(3)>0,f(9999)=3333$.

Determine $f(1982)$.

src_imho_1982__Q01

Triangle reflections and concurrent lines via angle bisector

A non-isosceles triangle $A_1{A}_2{A}_3$ is given with sides $a_1,a_2,a_3$ ($a_i$ is the side opposite $A_i$). For all $i=1,2,3$, $M_i$ is the midpoint of side $a_i$, and $T_i$ is the point where the incircle touches side $a_i$. Denote by $S_i$ the reflection of $T_i$ in the interior bisector of angle $A_i$. Prove that the lines $M_1{S}_1$, $M_2{S}_2$, $M_3{S}_3$ are concurrent.

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

Triangle reflections and concurrent lines by angle bisector

A non-isosceles triangle $A_1{A}_2{A}_3$ is given with sides $a_1,a_2,a_3$ ($a_i$ is the opposite side $A_i$). For all $i=1,2,3$, $M_i$ is the midpoint of side $a_i$, and $T_i$ is the point where the incircle touches side $a_i$. Denote by $S_i$ the reflection of $T_i$ in the interior bisector of angle $A_i$. Prove that the lines $M_1{S}_1$, $M_2{S}_2$, $M_3{S}_3$ are concurrent.

src_imho_1982__Q02

Infinite sequence of positive reals: prove lower bound and find sequence with upper bound

Consider the infinite sequences $\{x_i\}$ of positive real numbers with the following properties:

(a) Prove that for every such sequence there is an $n\ge 1$ such that $\frac{x_0^2}{x_1}+\frac{x_1^2}{x_2}+\cdots +\frac{x_{n-1}^2}{x_n}\ge \mathrm{3.999.}$ (b) Find such a sequence for which $\frac{x_0^2}{x_1}+\frac{x_1^2}{x_2}+\cdots +\frac{x_{n-1}^2}{x_n}<4$ for all $n\ge 1$.

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

Infinite sequence of positive reals: prove lower bound and find sequence with upper bound

Consider the infinite sequences $\{x_i\}$ of positive real numbers with the following properties: $x_0=1,\text{ and for all }i\ge 0,x_{i+1}\le x_i.$ (a) Prove that for every such sequence there is an $n\ge 1$ such that $\frac{x_0^2}{x_1}+\frac{x_1^2}{x_2}+\cdots +\frac{x_{n-1}^2}{x_n}\ge \mathrm{3.999.}$ (b) Find such a sequence for which $\frac{x_0^2}{x_1}+\frac{x_1^2}{x_2}+\cdots +\frac{x_{n-1}^2}{x_n}<4$ for all $n\ge 1$.

src_imho_1982__Q03

Prove x^2 - 3xy + y^2 = n has no integer solutions when n = 2^(2891)

Prove that if $n$ is a positive integer such that the equation $x^2-3xy+y^2=n$ has a solution in integers $(x,y)$, then it has at least three such solutions. Show that the equation has no solution in integers when $n=2^{2891}$.

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

*Prove x^2 - 3xy + y^2 = n has no integer solutions when n = 2^(2891) *

Prove that if $n$ is a positive integer such that the equation $x^2-3xy+y^2=n$ has a solution in integers $(x,y)$, then it has at least three such solutions. Show that the equation has no solution in integers when $n=2^{2891}$.

src_imho_1982__Q04

Regular hexagon diagonal ratio and collinearity of B, M, N

The diagonals $AC$ and $CE$ of the regular hexagon $ABCDEF$ are divided by the inner points $M$ and $N$ respectively, so that $\frac{AM}{AC}=\frac{CN}{CE}=r.$ Determine $r$ if $B$, $M$, $N$ are collinear.

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

Regular hexagon diagonal ratio and collinearity of B, M, N

The diagonals $AC$ and $CE$ of the regular hexagon $ABCDEF$ are divided by the inner points $M$ and $N$ respectively, so that $\frac{AM}{AC}=\frac{CN}{CE}=r.$ Determine $r$ if $B$, $M$, $N$ are collinear.

src_imho_1982__Q05

Path in square covering boundary within distance 1/2, prove length ≥ 198

Let $S$ be a square with sides of length 100, and let $L$ be a path within $S$ which does not meet itself and which is composed of line segments $A_0{A}_1,A_1{A}_2,\cdots ,A_{n-1}{A}_n$ with $A_0\ne A_n$. Suppose that for every point $P$ of the boundary of $S$ there is a point of $L$ at a distance from $P$ not greater than $1/2$. Prove that there are two points $X$ and $Y$ in $L$ such that the distance between $X$ and $Y$ is not greater than $1$, and the length of that part of $L$ which lies between $X$ and $Y$ is not smaller than $198$.

Topic: Combinatoria, Geometria piana Metodo: Estremalità, Principio dei cassetti, Invarianti Abilita: Ragionamento geometrico, Modellizzazione, Stima, Astrazione Area: Combinatoria, Logica e Probabilita, Geometria Fonte: apri PDF p.1

Path in square covering boundary within distance 1/2, length tests ≥ 198

Let $S$ be a square with sides of length 100, and let $L$ be a path within $S$ which does not meet itself and which is composed of line segments $A_0{A}_1,A_1{A}_2,\cdots ,A_{n-1}{A}_n$ with $A_0\ne A_n$. Suppose that for every point $P$ of the boundary of $S$ there is a point of $L$ at a distance from $P$ not greater than $1/2$. Prove that there are two points $X$ and $Y$ in $L$ such that the distance between $X$ and $Y$ is not greater than $1$, and the length of that part of $L$ which lies between $X$ and $Y$ is not smaller than $198$.

src_imho_1982__Q06