IMO 1978 (XX International Mathematical Olympiad)

Find m,n with equal last three digits of 78^m and 78^n

$m$ and $n$ are natural numbers with $1\le m<n$. In their decimal representations, the last three digits of $1978^m$ are equal, respectively, to the last three digits of $1978^n$. Find $m$ and $n$ such that $m+n$ has its least value.

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

Find m,n with equal last three digits of 78^m and 78

$m$ and $n$ are natural numbers with $1\le m<n$. In their decimal representations, the last three digits of $1978^m$ are equal, respectively, to the last three digits of $1978^n$. Find $m$ and $n$ such that $m+n$ has its least value.

src_imho_1978__Q01

Three mutually perpendicular rays from point P on a sphere

$P$ is a given point on a sphere. Three mutually perpendicular rays from $P$ intersect the sphere at points $U$, $V$, $W$. Prove that for all such triples of rays, the plane of triangle $UVW$ passes through a fixed point, and find the locus of $Q$ for all such triangles $UVW$.

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

Three mutually perpendicular rays from point P on a sphere

$P$ is a given point on a sphere. Three mutually perpendicular rays from $P$ intersect the sphere at points $U$, $V$, $W$. Prove that for all such triples of rays, the plane of triangle $UVW$ passes through a fixed point, and find the locus of $Q$ for all such triangles $UVW$.

src_imho_1978__Q02

Set of integers split into two sequences; find f(240)

The set of all positive integers is the union of two disjoint subsets $\{f(1),f(2),\dots ,f(n),\dots \}$, $\{g(1),g(2),\dots ,g(n),\dots \}$, where $f(1)<f(2)<\cdots <f(n)<\cdots ,$ $g(1)<g(2)<\cdots <g(n)<\cdots ,$ and $g(n)=f(f(n))+1$ for all $n\ge 1$. Determine $f(240)$.

Topic: Teoria dei Numeri, Combinatoria Metodo: Ricorsione, Induzione, Invarianti Abilita: Riconoscimento di pattern, Lettura attenta, Ragionamento geometrico Area: Aritmetica e Teoria dei Numeri, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

*Set of integers split into two sequences; find f(240) *

The set of all positive integers is the union of two disjoint subsets $\{f(1),f(2),\dots ,f(n),\dots \}$, $\{g(1),g(2),\dots ,g(n),\dots \}$, where $f(1)<f(2)<\cdots <f(n)<\cdots ,$ $g(1)<g(2)<\cdots <g(n)<\cdots ,$ and $g(n)=f(f(n))+1$ for all $n\ge 1$. Determine $f(240)$.

src_imho_1978__Q03

Incircle tangent to AB at D; midpoint of PQ is center of incircle

In triangle $ABC$, $AB=AC$. A circle is tangent internally to the circumcircle of triangle $ABC$ and also to sides $AB$, $AC$ at $P$, $Q$ respectively. Prove that the midpoint of segment $PQ$ is the center of the incircle of triangle $ABC$.

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

Incircle tangent to AB at D; midpoint of PQ is center of incircle

In the $ABC$ triangle, $AB=AC$. A circle is tangent internally to the circumcircle of triangle $ABC$ and also to sides $AB$, $AC$ at $P$, $Q$ respectively. Prove that the midpoint of segment $PQ$ is the center of the incircle of triangle $ABC$.

src_imho_1978__Q04

Sequence of distinct positive integers; sum inequality

Let $\{a_k\}$ $(k=1,2,3,\dots ,n,\dots )$ be a sequence of distinct positive integers. Prove that for all natural numbers $n$, ${\sum }_{k=1}^n\frac{a_k}{k^2}\ge {\sum }_{k=1}^n\frac{1}{k}.$

Topic: Disuguaglianze, Teoria dei Numeri Metodo: Disuguaglianze, Induzione, Principio dei cassetti Abilita: Manipolazione algebrica, Stima, Ragionamento geometrico Area: Algebra e Analisi, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Sequence of distinct positive integers; sum inequality

Let $\{a_k\}$ $(k=1,2,3,\dots ,n,\dots )$ be a sequence of distinct positive integers. Prove that for all natural numbers

src_imho_1978__Q05

International society; member whose number equals sum or double

An international society has its members from six different countries. The list of members contains $1978$ names, numbered $1,2,\dots ,1978$. Prove that there is at least one member whose number is the sum of the numbers of two members from his own country, or twice as large as the number of one member from his own country.

Topic: Combinatoria, Teoria dei Numeri Metodo: Principio dei cassetti, Estremalità, Induzione Abilita: Ragionamento geometrico, Conteggio sistematico, Lettura attenta Area: Combinatoria, Logica e Probabilita, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

International society; member whose number is equal to sum or double

An international society has its members from six different countries. The list of members contains $1978$ names, numbered $1,2,\dots ,1978$. Prove that there is at least one member whose number is the sum of the numbers of two members from his own country, or twice as large as the number of one member from his own country.

src_imho_1978__Q06