IMO 1971 (Thirteenth International Olympiad)

Symmetric product inequality true only for n=3,5

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

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: Disuguaglianze, Algebra Metodo: Disuguaglianze classiche Area: Algebra e Analisi Fonte: apri PDF p.1

Symmetric product inequality true only for n=3,5

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

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_imo_1971_all__Q01

Two of nine translated polyhedra share interior 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 Metodo: Principio dei cassetti Area: Geometria Fonte: apri PDF p.1

Two of nine translated polyhedra share interior 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.

src_imo_1971_all__Q02

Infinite pairwise-coprime subset of numbers 2^k-3

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 Metodo: congruenze Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Infinite pairwise-coprime subset of numbers 2^k-3

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_imo_1971_all__Q03

Minimal closed polygonal path on tetrahedron edges

All the faces of tetrahedron $ABCD$ are acute-angled triangles. We consider all closed polygonal paths of the form $XYZTX$ 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 are 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 $\alpha =\angle BAC+\angle CAD+\angle ADB$.

Topic: Geometria solida Area: Geometria Fonte: apri PDF p.1

Minimal closed polygonal path on tetrahedron edges

All the faces of tetrahedron $ABCD$ are acute-angled triangles. We consider all closed polygonal paths of the form $XYZTX$ 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 are 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 $\alpha =\angle BAC+\angle CAD+\angle ADB$.

src_imo_1971_all__Q04

Point set where every point has m points at unit distance

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 Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Point set where every point has m points at unit distance

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_imo_1971_all__Q05

Nonnegative-integer matrix with zero-sum condition has sum>=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: Combinatoria Metodo: Doppio conteggio Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Nonnegative-integer matrix with zero-sum condition has sum>=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_imo_1971_all__Q06