IMO 1977 (XIX International Mathematical Olympiad)

Midpoints of segments in square form dodecagon vertices

Equilateral triangles $ABK$, $BCL$, $CDM$, $DAN$ are constructed inside the square $ABCD$. Prove that the midpoints of the four segments $KL$, $LM$, $MN$, $NK$ and the midpoints of the eight segments $AB$, $BK$, $BL$, $BC$, $CM$, $DM$, $DN$, $AN$ are the twelve vertices of a regular dodecagon.

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

Midpoints of segments in square form dodecagon vertices

Equilateral triangles $ABK$, $BCL$, $CDM$, $DAN$ are constructed inside the square $ABCD$. Prove that the midpoints of the four segments $KL$, $LM$, $MN$, $NK$ and the midpoints of the eight segments $AB$, $BK$, $BL$, $CM$, $DM$, $DN$, $AN$ are the twelve vertices of a regular dodecagon.

src_imho_1977__Q01

Finite sequence: max terms negative with seven successive sum negative

In a finite sequence of real numbers the sum of any seven successive terms is negative, and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.

Topic: Algebra, Combinatoria Metodo: Casework, Estremalità Abilita: Ragionamento geometrico, Manipolazione algebrica, Lettura attenta Area: Algebra e Analisi, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Finite sequence: max terms negative with seven successive sum negative

In a finite sequence of real numbers the sum of any seven successive terms is negative, and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.

src_imho_1977__Q02

Indecomposable numbers in set $V_k$; prove $p=m$

Let $n$ be a given integer $>2$, and let $V_k$ be the set of integers $1+kn$, where $k=1,2,\dots$ A number $m\in V_k$ is called $\text{indecomposable}$ in $V_k$ if there do not exist numbers $p,q\in V_k$ such that $pq=m$. Prove that there exists a number $r\in V_k$ that can be expressed as the product of elements indecomposable in $V_k$ in more than one way. (Products which differ only in the order of their factors will be considered the same.)

Topic: Teoria dei Numeri Metodo: Congruenze, Fattorizzazione, Casework Abilita: Manipolazione algebrica, Riconoscimento di pattern, Astrazione Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Indecomposable numbers in set $V_k$; proofs $p=m$

Let $n$ be a given integer $>2$, and let $V_k$ be the set of integers $1+kn$, where $k=1,2,\dots$ A number $m\in V_k$ is called $\text{indecomposable}$ in $V_k$ if there do not exist numbers $p,q\in V_k$ such that $pq=m$. Prove that there exists a number $r\in V_k$ that can be expressed as the product of elements indecomposable in $V_k$ in more than one way. (Products which differ only in the order of their factors will be considered the same.)

src_imho_1977__Q03

Four real constants; prove $f(\theta )\ge 0$ implies $A^2+B^2\le 1$

Four real constants $a$, $b$, $A$, $B$ are given, and $f(\theta )=1-a\cos \theta -b\sin \theta -A\cos 2\theta -B\sin 2\theta .$ Prove that if $f(\theta )\ge 0$ for all real $\theta$, then $a^2+b^2\le 2$ and $A^2+B^2\le 1$.

Topic: Disuguaglianze, Trigonometria Metodo: Disuguaglianze, Trigonometria Abilita: Manipolazione algebrica, Stima, Lettura attenta Area: Algebra e Analisi, Geometria Fonte: apri PDF p.1

Four real constants; proofs $f(\theta )\ge 0$ implies $A^2+B^2\le 1$

Four real constants $a$, $b$, $A$, $B$ are given, and $f(\theta )=1-a\cos \theta -b\sin \theta -A\cos 2\theta -B\sin 2\theta .$ Prove that if $f(\theta )\ge 0$ for all real $\theta$, then $a^2+b^2\le 2$ and $A^2+B^2\le 1$.

src_imho_1977__Q04

Pairs $(a,b)$ with $a^2+b^2$ divisible by $a+b$; remainder $r=1977$

Let $a$ and $b$ be positive integers. When $a^2+b^2$ is divided by $a+b$, the quotient is $q$ and the remainder is $r$. Prove that there are infinitely many pairs $(a,b)$ such that $q^2+r=1977$.

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

Parts $(a,b)$ with $a^2+b^2$ divisible by $a+b$; remainder $r=1977$*

Let $a$ and $b$ be positive integers. When $a^2+b^2$ is divided by $a+b$, the quotient is $q$ and the remainder is $r$. Prove that there are infinitely many pairs $(a,b)$ such that $q^2+r=1977$.

src_imho_1977__Q05

Function on positive integers with $f(n+1)>f(f(n))$; prove $f(n)=n$

Let $f(n)$ be a function defined on the set of all positive integers and having all its values in the same set. Prove that if $f(n+1)>f(f(n))$ for each positive integer $n$, then $f(n)=n$ for each positive integer $n$.

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

Function on positive integers with $f(n+1)>f(f(n))$; proofs $f(n)=n$

Let $f(n)$ be a function defined on the set of all positive integers and having all its values in the same set. Prove that if $f(n+1)>f(f(n))$ for each positive integer $n$, then $f(n)=n$ for each positive integer $n$.

src_imho_1977__Q06