IMO 1984 (25th International Mathematical Olympiad)

Bound xy+yz+zx-2xyz for nonnegative reals summing to 1

Prove that $0\le yz+zx+xy-2xyz\le \frac{7}{27}$, where $x$, $y$ and $z$ are non-negative real numbers for which $x+y+z=1$.

Topic: Disuguaglianze Metodo: Disuguaglianze classiche Area: Algebra e Analisi Fonte: apri PDF p.1

Bound xy+yz+zx-2xyz for nonnegative reals summing to 1

Prove that $0\le yz+zx+xy-2xyz\le \frac{7}{27}$, where $x$, $y$ and $z$ are non-negative real numbers for which $x+y+z=1$.

src_imo_1984__Q01

Find positive integers with divisibility conditions modulo 7 powers

Find one pair of positive integers $a$ and $b$ such that:

(i) $ab(a+b)$ is not divisible by $7$;

(ii) $(a+b{)}^7-a^7-b^7$ is divisible by $7^7$.

Justify your answer.

Topic: Teoria dei Numeri Metodo: congruenze Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Find positive integers with divisibility conditions module 7 powers

Find one pair of positive integers $a$ and $b$ such that:

(i) $ab(a+b)$ is not divisible by $7$;

(ii) $(a+b{)}^7-a^7-b^7$ is divisible by $7^7$.

Justify your answer.

src_imo_1984__Q02

point with same color on its circle

In the plane two different points $O$ and $A$ are given. For each point $X$ of the plane, other than $O$, denote by $a(X)$ the measure of the angle between $OA$ and $OX$ in radians, counterclockwise from $OA$ ($0\le a(X)<2\pi$). Let $C(X)$ be the circle with center $O$ and radius of length $OX+a(X)/OX$. Each point of the plane is colored by one of a finite number of colors. Prove that there exists a point $Y$ for which $a(Y)>0$ such that its color appears on the circumference of the circle $C(Y)$.

Topic: Combinatoria Metodo: Principio dei cassetti Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

point with same color on its circle

In the plane two different points $O$ and $A$ are given. For each point $X$ of the plane, other than $O$, denoted by $a(X)$ the measure of the angle between $OA$ and $OX$ in radians, counterclockwise from $OA$ ($0\le a(X)<2\pi$). Let $C(X)$ be the circle with center $O$ and radius of length $OX+a(X)/OX$. Each point of the plane is colored by one of a finite number of colors. Prove that there exists a point $Y$ for which $a(Y)>0$ such that its color appears on the circumference of the circle $C(Y)$.

src_imo_1984__Q03

Tangent circle on AB diameter iff BC parallel AD

Let $ABCD$ be a convex quadrilateral such that the line $CD$ is a tangent to the circle on $AB$ as diameter. Prove that the line $AB$ is a tangent to the circle on $CD$ as diameter if and only if the lines $BC$ and $AD$ are parallel.

Topic: Geometria piana Abilita: Ragionamento geometrico Area: Geometria Fonte: apri PDF p.1

Tangent circle on AB diameter iff BC parallel AD

Let $ABCD$ be a convex quadrilateral such that the line $CD$ is a tangent to the circle on $AB$ as diameter. Prove that the line $AB$ is a tangent to the circle on $CD$ as diameter if and only if the lines $BC$ and $AD$ are parallel.

src_imo_1984__Q04

Bound diagonal-sum to perimeter ratio of convex polygon

Let $d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $n$ vertices ($n>3$), and let $p$ be its perimeter. Prove that $n-3<\frac{2d}{p}<\left[\frac{n}{2}\right]\left[\frac{n+1}{2}\right]-2,$ where $[x]$ denotes the greatest integer not exceeding $x$.

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

Bound diagonal-sum to perimeter ratio of convex polygon

Let $d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $n$ vertices ($n>3$), and let $p$ be its perimeter. Prove that $n-3<\frac{2d}{p}<\left[\frac{n}{2}\right]\left[\frac{n+1}{2}\right]-2,$ where $[x]$ denotes the greatest integer not exceeding $x$.

src_imo_1984__Q05

Odd integers with ad=bc and power-of-two sums imply a=1

Let $a$, $b$, $c$ and $d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.

Topic: Teoria dei Numeri Metodo: congruenze Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Odd integers with ad=bc and power-of-two sums imply a=1

Let $a$, $b$, $c$ and $d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.

src_imo_1984__Q06