IMO 1979 (XXI International Mathematical Olympiad)

Divisibility of a sum involving unit fractions

Let $p$ and $q$ be natural numbers such that $\frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots -\frac{1}{1318}+\frac{1}{1319}.$ Prove that $p$ is divisible by $1979$.

Topic: Teoria dei Numeri Metodo: Congruenze, Telescoping Abilita: Manipolazione algebrica, Ragionamento geometrico, Lettura attenta Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Divisibility of a sum involving unit fractions

Let $p$ and $q$ be natural numbers such that $\frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots -\frac{1}{1318}+\frac{1}{1319}.$ Prove that $p$ is divisible by $1979$.

src_imho_1979__Q01

Two-colored pentagons on top and bottom faces of prism

A prism with pentagons $A_1{A}_2{A}_3{A}_4{A}_5$ and $B_1{B}_2{B}_3{B}_4{B}_5$ as top and bottom faces is given. Each of the two pentagons and each of the line-segments $A_i{B}_i$ for all $i=1,\dots ,5$ is colored either red or green. Every triangle whose vertices are vertices of the prism and whose sides are all colored segments has all its sides the same color. Show that all 10 sides of the top and bottom faces are the same color.

Topic: Combinatoria Metodo: Colorazione, Casework Abilita: Ragionamento geometrico, Casework accurato, Lettura attenta Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Two-colored pentagons on top and bottom faces of prism

A prism with pentagons $A_1{A}_2{A}_3{A}_4{A}_5$ and $B_1{B}_2{B}_3{B}_4{B}_5$ as top and bottom faces is given. Each of the two pentagons and each of the line segments $A_i{B}_i$ for all $i=1,\dots ,5$ is colored either red or green. Every triangle whose vertices are vertices of the prism and whose sides are all colored segments has all its sides the same color. Show that all 10 sides of the top and bottom faces are the same color.

src_imho_1979__Q02

Two circles with common tangent lines meet at intersection point

Two circles in a plane intersect. Let $A$ be one of the points of intersection. Starting simultaneously from $A$, two points move with constant speeds, each point travelling along its own circle in the same sense. The two points return to $A$ simultaneously after each has made exactly one full revolution. Prove that there is a fixed point $P$ in the plane such that, at any time, the distances from $P$ to the two moving points are equal.

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

Two circles with common tangent lines meet at intersection point

Two circles in a plane intersect. Let $A$ be one of the points of intersection. Starting simultaneously from $A$, two points move with constant speeds, each point traveling along its own circle in the same sense. The two points return to $A$ simultaneously after each has made exactly one full revolution. Prove that there is a fixed point $P$ in the plane such that, at any time, the distances from $P$ to the two moving points are equal.

src_imho_1979__Q03

All real numbers in pi make distances equal to fixed point

Given a plane $\pi$, a point $P$ in this plane and a point $Q$ not in $\pi$. Find all points $R$ in $\pi$ such that $\frac{QP+PR}{QR}$ is a maximum.

Topic: Geometria analitica, Geometria solida Metodo: Disuguaglianze, Estremalità Abilita: Ragionamento geometrico, Manipolazione algebrica, Modellizzazione Area: Geometria Fonte: apri PDF p.1

All real numbers in pi make distances equal to fixed point

Given a plane $\pi$, a point $P$ in this plane and a point $Q$ not in $\pi$. Find all points $R$ in $\pi$ such that $\frac{QP+PR}{QR}$ is a maximum.

src_imho_1979__Q04

Non-negative reals satisfying two symmetric sum equations

Find all real numbers $x_1,x_2,x_3,x_4,x_5$ satisfying the relations ${\sum }_{k=1}^{5}k{x}_k=a,{\sum }_{k=1}^5{k}^3{x}_k=a^2,{\sum }_{k=1}^5{k}^5{x}_k=a^3$ for some real number $a$.

Topic: Algebra Metodo: Disuguaglianze, Fattorizzazione Abilita: Manipolazione algebrica, Riconoscimento di pattern, Lettura attenta Area: Algebra e Analisi Fonte: apri PDF p.1

Non-negative reals satisfying two symmetric sum equations

Find all real numbers $x_1,x_2,x_3,x_4,x_5$ satisfying the relations ${\sum }_{k=1}^{5}k{x}_k=a,{\sum }_{k=1}^5{k}^3{x}_k=a^2,{\sum }_{k=1}^5{k}^5{x}_k=a^3$ for some real number $a$.

src_imho_1979__Q05

Frog on regular octagon: count paths ending at opposite vertex

Let $A$ and $E$ be opposite vertices of a regular octagon. A frog starts jumping at vertex $A$. From any vertex of the octagon except $E$, it may jump to either of the two adjacent vertices. When it reaches vertex $E$, the frog stops and stays there. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E$. Prove that $a_{2n-1}=0,a_{2n}=\frac{1}{\sqrt{2}}\left(x^{n-1}-y^{n-1}\right),n=1,2,3,\dots ,$ where $x=2+\sqrt{2}$ and $y=2-\sqrt{2}$.

\textit{Note.} A path of $n$ jumps is a sequence of vertices $(P_0,P_1,\dots ,P_n)$ such that \begin{itemize} \item[(i)] $P_0=A$, $P_n=E$; \item[(ii)] for every $i$, $0\le i\le n-1$, $P_i$ is distinct from $E$; \item[(iii)] for every $i$, $0\le i\le n-1$, $P_i$ and $P_{i+1}$ are adjacent. \end{itemize}

Topic: Combinatoria Metodo: Ricorsione, Induzione, Grafi Abilita: Riconoscimento di pattern, Conteggio sistematico, Manipolazione algebrica, Modellizzazione Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Frog on regular octagon: count paths ending at opposite vertex

Let $A$ and $E$ be opposite vertices of a regular octagon. A frog starts jumping at vertex. From any vertex of the octagon except $E$, it may jump to either of the two adjacent vertices. When it reaches vertex, the frog stops and stays there. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E$. Prove that $a_{2n-1}=0,a_{2n}=\frac{1}{\sqrt{2}}\left(x^{n-1}-y^{n-1}\right),n=1,2,3,\dots ,$ where $x=2+\sqrt{2}$ and $y=2-\sqrt{2}$.

\textit{Note.} A path of $n$ jumps is a sequence of vertices $(P_0,P_1,\dots ,P_n)$ such that \begin{itemize} \item[(i)] $P_0=A$, $P_n=E$; \item[(ii)] for every $i$, $0\le i\le n-1$, $P_i$ is distinct from $E$; \item[(iii)] for every $i$, $0\le i\le n-1$, $P_i$ and $P_{i+1}$ are adjacent. I’m going to tell you.

src_imho_1979__Q06