IMO 1984 (XXV International Mathematical Olympiad)

Prove double inequality for symmetric expression with x+y+z=1

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

Topic: Disuguaglianze, Algebra Metodo: Disuguaglianze, Estremalità, Simmetria Abilita: Manipolazione algebrica, Stima Area: Algebra e Analisi Fonte: apri PDF p.1

Prove double inequality for symmetric expression with x+y+z=1*

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

src_imho_1984__Q01

Find positive integers a, b with ab(a+b) not divisible by 7 but (a+b)^7−a^7−b^7 divisible by 7^7

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, Fattorizzazione Abilita: Manipolazione algebrica, Riconoscimento di pattern, Lettura attenta Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Find positive integers a, b with ab(a+b) not divisible by 7 but (a+b)^7−a^7−b^7 divisible by 7^7

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_imho_1984__Q02

Finitely-colored plane: some color touches every circle C(X)

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 $OX$ (counterclockwise from $OA$ to $OX$, $0\le a(X)<2\pi$). Let $C(X)$ be the circle with center $O$ and radius of length $OX\cdot a(X)/OX$. Each point of the plane is colored by one of a finite number of colors. Prove that there exists a point $Y$ such that its color appears on the circumference of the circle $C(Y)$.

Topic: Combinatoria, Geometria piana Metodo: Colorazione, Principio dei cassetti, Estremalità Abilita: Astrazione, Ragionamento geometrico, Riconoscimento di pattern Area: Combinatoria, Logica e Probabilita, Geometria Fonte: apri PDF p.1

Finely-colored plane: some color touches every 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 $OX$ (counterclockwise from $OA$ to $OX$, $0\le a(X)<2\pi$). Let $C(X)$ be the circle with center $O$ and radius of length $OX\cdot a(X)/OX$. Each point of the plane is colored by one of a finite number of colors. Prove that there exists a point $Y$ such that its color appears on the circumference of the circle $C(Y)$.

src_imho_1984__Q03

Convex quadrilateral ABCD: CD tangent to circle on AB iff BC ∥ 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 Metodo: Trigonometria, Coordinate Abilita: Ragionamento geometrico, Manipolazione algebrica, Lettura attenta Area: Geometria Fonte: apri PDF p.1

Convex quadrilateral ABCD: CD tangent to circle on AB iff BC 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_imho_1984__Q04

Diagonal sum vs perimeter double inequality for convex n-gon

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 $n-3<\frac{2d}{p}<\lfloor \frac{n}{2}\rfloor \cdot \frac{n+1}{2}-2,$ where $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$.

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

Diagonal sum vs perimeter double inequality for convex n-gon

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 $n-3<\frac{2d}{p}<\lfloor \frac{n}{2}\rfloor \cdot \frac{n+1}{2}-2,$ where $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$.

src_imho_1984__Q05

Odd integers with ad=bc, a+d=2^k, b+c=2^m imply a=1

Let $a,b,c,d$ 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, Algebra Metodo: Congruenze, Fattorizzazione, Estremalità Abilita: Manipolazione algebrica, Lettura attenta, Ragionamento geometrico Area: Aritmetica e Teoria dei Numeri, Algebra e Analisi Fonte: apri PDF p.1

Odd integers with ad=bc, a+d=2^k, b+c=2^m imply a=1

Let $a,b,c,d$ 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_imho_1984__Q06