IMO 1971 (XIII International Mathematical Olympiad)

Prove inequality for arbitrary real numbers, true for n=3,5 false otherwise

Prove that the following assertion is true for $n=3$ and $n=5$, and that it is false for every other $n>2$.

If $a_1,a_2,\dots ,a_n$ are arbitrary real numbers, then $(a_1-a_2)(a_1-a_3)\cdots (a_1-a_n)+(a_2-a_1)(a_2-a_3)\cdots (a_2-a_n)+\cdots +(a_n-a_1)(a_n-a_2)\cdots (a_n-a_{n-1})\ge 0$

Topic: Algebra, Disuguaglianze Metodo: Casework, Estremalità Abilita: Manipolazione algebrica, Lettura attenta, Casework accurato Area: Algebra e Analisi Fonte: apri PDF p.1

Prove inequality for arbitrary real numbers, true for n=3,5 false otherwise

Prove that the following statement is true for $n=3$ and $n=5$, and that it is false for every other $n>2$.

If $a_1,a_2,\dots ,a_n$ are arbitrary real numbers, then $(a_1-a_2)(a_1-a_3)\cdots (a_1-a_n)+(a_2-a_1)(a_2-a_3)\cdots (a_2-a_n)+\cdots +(a_n-a_1)(a_n-a_2)\cdots (a_n-a_{n-1})\ge 0$

src_imho_1971__Q01

Nine-vertex convex polyhedra sharing a common point

Consider a convex polyhedron $P_1$ with nine vertices $A_1,A_2,\dots ,A_9$; let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves vertex $A_1$ to $A_i$ $(i=2,3,\dots ,9)$. Prove that at least two of the polyhedra $P_1,P_2,\dots ,P_9$ have an interior point in common.

Topic: Geometria solida, Combinatoria Metodo: Principio dei cassetti, Invarianti Abilita: Modellizzazione, Ragionamento geometrico, Astrazione Area: Geometria, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

The value of the underlying assets shall be the sum of the assets of the underlying assets of the underlying assets.

Consider a convex polyhedron $P_1$ with nine vertices $A_1,A_2,\dots ,A_9$; let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves vertex $A_1$ to $A_i$ $(i=2,3,\dots ,9)$. Prove that at least two of the polyhedra $P_1,P_2,\dots ,P_9$ have an interior point in common.

src_imho_1971__Q02

Integers of form 2^3k-2 contain infinite subset with pairwise coprime elements

Prove that the set of integers of the form $2^k-3$ $(k=2,3,\dots )$ contains an infinite subset in which every two members are relatively prime.

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

Integers of form 2^3k-2 contain infinite subset with pairwise coprime elements

Prove that the set of integers of the form $2^k-3$ $(k=2,3,\dots )$ contains an infinite subset in which every two members are relatively prime.

src_imho_1971__Q03

Acute-angled triangles and minimal geodesic on tetrahedron ABCD

All the faces of tetrahedron $ABCD$ are acute-angled triangles. We consider all closed polygonal paths of the form $XYZТX$ defined as follows: $X$ is a point on edge $AB$ distinct from $A$ and $B$; similarly, $Y$, $Z$, $T$ are interior points of edges $BC$, $CD$, $DA$, respectively. Prove:

(a) If $\angle DAB+\angle BCD\ne \angle ABC+\angle CDA$, then among the polygonal paths, there is none of minimal length.

(b) If $\angle DAB+\angle BCD=\angle ABC+\angle CDA$, then there are infinitely many shortest polygonal paths, their common length being $2AC\sin (\alpha /2)$, where $\alpha =\angle BAC+\angle CAD+\angle DAB$.

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

Acute-angled triangles and minimal geodesic on tetrahedron ABCD

All the faces of tetrahedron $ABCD$ are acute-angled triangles. We consider all closed polygonal paths of the form $XYZТX$ defined as follows: $X$ is a point on edge $AB$ distinct from $A$ and $B$; similarly, $Y$, $Z$, $T$ are interior points of edges $BC$, $CD$, $DA$, respectively. Proofs:

(a) If $\angle DAB+\angle BCD\ne \angle ABC+\angle CDA$, then among the polygonal paths, there is none of minimum length.

(b) If $\angle DAB+\angle BCD=\angle ABC+\angle CDA$, then there are infinitely many shortest polygonal paths, their common length being $2AC\sin (\alpha /2)$, where $\alpha =\angle BAC+\angle CAD+\angle DAB$.

src_imho_1971__Q04

Finite set S in plane with exactly n points at unit distance from each point

Prove that for every natural number $m$, there exists a finite set $S$ of points in a plane with the following property: for every point $A$ in $S$, there are exactly $m$ points in $S$ which are at unit distance from $A$.

Topic: Combinatoria, Geometria piana Metodo: method_casework, Induzione Abilita: Modellizzazione, Astrazione, Ragionamento geometrico Area: Combinatoria, Logica e Probabilita, Geometria Fonte: apri PDF p.1

Finite set S in plane with exactly n points at unit distance from each point

Prove that for every natural number $m$, there exists a finite set $S$ of points in a plane with the following property: for every point $A$ in $S$, there are exactly $m$ points in $S$ which are at unit distance from $A$.

src_imho_1971__Q05

Square matrix with row/column sums >= n implies sum of all elements >= n^2/2

Let $A=(a_{ij})$ $(i,j=1,2,\dots ,n)$ be a square matrix whose elements are non-negative integers. Suppose that whenever an element $a_{ij}=0$, the sum of the elements in the $i$th row and the $j$th column is $\ge n$. Prove that the sum of all the elements of the matrix is $\ge n^2/2$.

Topic: Algebra, Combinatoria, Disuguaglianze Metodo: Doppio conteggio, Disuguaglianze, Casework Abilita: Manipolazione algebrica, Conteggio sistematico, Lettura attenta Area: Algebra e Analisi, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Square matrix with row/column sums >= n implies sum of all elements >= n^2/2

Let $A=(a_{ij})$ $(i,j=1,2,\dots ,n)$ be a square matrix whose elements are non-negative integers. Suppose that whenever an element $a_{ij}=0$, the sum of the elements in the $i$th row and the $j$th column is $\ge n$. Prove that the sum of all the elements of the matrix is $\ge n^2/2$.

src_imho_1971__Q06