IMO 1967 (IX International Mathematical Olympiad)

Parallelogram with four unit circles covering it

Let $ABCD$ be a parallelogram with side lengths $AB=a$, $AD=1$, and $\angle BAD=\alpha$. If $△ABD$ is acute, prove that the four circles of radius 1 with centers $A$, $B$, $C$, $D$ cover the parallelogram if and only if $a\le \cos \alpha +\sqrt{3}\sin \alpha .$

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

Parallelogram with four unit circles covering it

Let $ABCD$ be a parallelogram with side lengths $AB=a$, $AD=1$, and $\angle BAD=\alpha$. If $△ABD$ is acute, prove that the four circles of radius 1 with centers $A$, $B$, $C$, $D$ cover the parallelogram if and only if $a\le \cos \alpha +\sqrt{3}\sin \alpha .$

src_imho_1967__Q01

Tetrahedron edge greater than 1 implies volume at most 1/8

Prove that if one and only one edge of a tetrahedron is greater than 1, then its volume is $\le 1/8$.

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

Tetrahedron edge greater than 1 implies volume at most 1/8

Prove that if one and only one edge of a tetrahedron is greater than 1, then its volume is $\le 1/8$.

src_imho_1967__Q02

Divisibility of product of consecutive integers by prime

Let $k$, $m$, $n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$. Let $c_s=s(s+1)$. Prove that the product $(c_{m+1}-c_k)(c_{m+2}-c_k)\cdots (c_{m+n}-c_k)$ is divisible by $c_1{c}_2\cdots c_n$.

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

Divisibility of product of consecutive integers by prime

Let $k$, $m$, $n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$. Let $c_s=s(s+1)$. Prove that the product $(c_{m+1}-c_k)(c_{m+2}-c_k)\cdots (c_{m+n}-c_k)$ is divisible by $c_1{c}_2\cdots c_n$.

src_imho_1967__Q03

Acute-angled triangles similar to ABC with maximum area

Let $A_0{B}_0{C}_0$ and $A_1{B}_1{C}_1$ be any two acute-angled triangles. Consider all triangles $ABC$ that are similar to $△A_1{B}_1{C}_1$ (so that vertices $A_1$, $B_1$, $C_1$ correspond to vertices $A$, $B$, $C$ respectively) and circumscribed about triangle $A_0{B}_0{C}_0$ (where $A_0$ lies on $BC$, $B_0$ on $CA$, and $C_0$ on $AB$). Of all such possible triangles, determine the one with maximum area, and construct it.

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

Acute-angled triangles similar to ABC with maximum area

Let $A_0{B}_0{C}_0$ and $A_1{B}_1{C}_1$ be any two acute-angled triangles. Consider all $ABC$ triangles that are similar to $△A_1{B}_1{C}_1$ (so that vertices $A_1$, $B_1$, $C_1$ correspond to vertices $A$, $B$, $C$ respectively) and circumscribed about triangle $A_0{B}_0{C}_0$ (where $A_0$ lies on $BC$, $B_0$ on $CA$, and $C_0$ on $AB$). Of all such possible triangles, determine the one with maximum area, and construct it.

src_imho_1967__Q04

Sequence of power sums vanishing infinitely often

Consider the sequence $\{c_n\}$, where $c_1=a_1+a_2+\cdots +a_k,$ $c_2=a_1^2+a_2^2+\cdots +a_k^2,$ $⋮$ $c_n=a_1^n+a_2^n+\cdots +a_k^n,$ in which $a_1,a_2,\dots ,a_k$ are real numbers not all equal to zero. Suppose that an infinite number of terms of the sequence $\{c_n\}$ are equal to zero. Find all natural numbers $n$ for which $c_n=0$.

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

Sequence of power sums vanishing infinitely often

Consider the sequence $\{c_n\}$, where $c_1=a_1+a_2+\cdots +a_k,$ $c_2=a_1^2+a_2^2+\cdots +a_k^2,$ $⋮$ $c_n=a_1^n+a_2^n+\cdots +a_k^n,$ in which $a_1,a_2,\dots ,a_k$ are real numbers not all equal to zero. Suppose that an infinite number of terms of the sequence $\{c_n\}$ are equal to zero. Find all natural numbers $n$ for which $c_n=0$.

src_imho_1967__Q05

Medal distribution over n days with 1/7 rule

In a sports contest, there were $m$ medals awarded on $n$ successive days ($n>1$). On the first day, one medal and $1/7$ of the remaining $m-1$ medals were awarded. On the second day, two medals and $1/7$ of the now remaining medals were awarded; and so on. On the $n$-th and last day, the remaining $n$ medals were awarded. How many days did the contest last, and how many medals were awarded altogether?

Topic: Teoria dei Numeri, Algebra Metodo: Ricorsione, Induzione, Fattorizzazione Abilita: Manipolazione algebrica, Modellizzazione, Lettura attenta, Conteggio sistematico Area: Aritmetica e Teoria dei Numeri, Algebra e Analisi Fonte: apri PDF p.1

Medal distribution over n days with 1/7 rule

In a sports competition, there were $m$ medals awarded on $n$ subsequent days ($n>1$). On the first day, one medal and $1/7$ of the remaining $m-1$ medals were awarded. On the second day, two medals and $1/7$ of the now remaining medals were awarded; and so on. On the $n$-th and last day, the remaining $n$ medals were awarded. How many days did the contest last, and how many medals were awarded altogether?

src_imho_1967__Q06