IMO 1978 (Twentieth International Olympiad)

Minimize m+n with equal last three digits of 1978^m,1978^n

Let $m$ and $n$ be 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 Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Minimize m+n with equal last three digits of 1978^m,1978

Let $m$ and $n$ be 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_imo_1978_all__Q01

Locus of opposite vertex Q of perpendicular rays in sphere

$P$ is a given point inside a given sphere. Three mutually perpendicular rays from $P$ intersect the sphere at points $U$, $V$, and $W$; $Q$ denotes the vertex diagonally opposite to $P$ in the parallelepiped determined by $PU$, $PV$, and $PW$. Find the locus of $Q$ for all such triads of rays from $P$.

Topic: Geometria solida Metodo: Metodo delle coordinate Area: Geometria Fonte: apri PDF p.1

Locus of opposite vertex Q of perpendicular rays in sphere

$P$ is a given point inside a given sphere. Three mutually perpendicular rays from $P$ intersect the sphere at points $U$, $V$, and $W$; $Q$ denotes the vertex diagonally opposite to $P$ in the parallelpiped determined by $PU$, $PV$, and $PW$. Find the locus of $Q$ for all such triads of rays from $P$.

src_imo_1978_all__Q02

find f(240)

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

Topic: successioni, Teoria dei Numeri Metodo: manipolazione algebrica Area: Algebra e Analisi, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

The Commission shall adopt delegated acts in accordance with Article 21 of the Financial Regulation.

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

src_imo_1978_all__Q03

Midpoint of tangency chord PQ is incircle center

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: Metodo delle coordinate Abilita: Ragionamento geometrico Area: Geometria Fonte: apri PDF p.1

Midpoint of tangency chord PQ is incircle center

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_imo_1978_all__Q04

sum a_k/k^2 >= sum 1/k

Let $\{a_k\}$ ($k=1,2,3,\dots$) 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 Metodo: Disuguaglianze classiche Area: Algebra e Analisi Fonte: apri PDF p.1

sum a_k/k^2 >= sum 1/k

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

src_imo_1978_all__Q05

Member whose number is sum/double of same-country members

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 Metodo: Principio dei cassetti Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Member whose number is sum/double of same-country members

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_imo_1978_all__Q06