\hfill \thepage} %} \input{tcilatex} \begin{document} \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 $\geq n^{2}/2.$ \end{document}