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\section{Thirteenth International Olympiad, 1971}

\subsection{1971/1.}

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 $a_{1},a_{2},...,a_{n}$ are arbitrary real numbers, then

$(a_{1}-a_{2})(a_{1}-a_{3})\cdots
(a_{1}-a_{n})+(a_{2}-a_{1})(a_{2}-a_{3})\cdots (a_{2}-a_{n})$

$+\cdots +(a_{n}-a_{1})(a_{n}-a_{2})\cdots (a_{n}-a_{n-1})\geq 0$

\subsection{1971/2.}

Consider a convex polyhedron $P_{1}$ with nine vertices $%
A_{1}A_{2},...,A_{9};$ let $P_{i}$ be the polyhedron obtained from $P_{1}$
by a translation that moves vertex $A_{1}$ to $A_{i}(i=2,3,...,9).$ Prove
that at least two of the polyhedra $P_{1},P_{2},...,P_{9}$ have an interior
point in common.

\subsection{1971/3.}

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

\subsection{1971/4.}

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 $BCCD,DA,$ respectively. Prove:

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

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

\subsection{1971/5.}

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 $A.$

\subsection{1971/6.}

Let $A=(a_{ij})(i,j=1,2,...,n)$ 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 $\geq n$. Prove
that the sum of all the elements of the matrix is $\geq n^{2}/2.$

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