\hfill \thepage} %} \input{tcilatex} \begin{document} \section{Seventh Internatioaal Olympiad, 1965} \subsection{1965/1.} Determine all values $x$ in the interval $0\leq x\leq 2\pi $ which satisfy the inequality \[ 2\cos x\leq \left| \sqrt{1+\sin 2x}-\sqrt{1-\sin 2x}\right| \leq \sqrt{2}. \] \subsection{1965/2.} Consider the system of equations \begin{eqnarray*} a_{11}x_{1}+a_{12}x_{2}+a_{13}x_{3} &=&0 \\ a_{21}x_{1}+a_{22}x_{2}+a_{23}x_{3} &=&0 \\ a_{31}x_{1}+a_{32}x_{2}+a_{33}x_{3} &=&0 \end{eqnarray*} with unknowns $x_{1},x_{2},x_{3}$. The coefficients satisfy the conditions: (a) $a_{11},a_{22},a_{33}$ are positive numbers; (b) the remaining coefficients are negative numbers; (c) in each equation, the sum of the coefficients is positive. Prove that the given system has only the solution $x_{1}=x_{2}=x_{3}=0$. \subsection{1965/3.} Given the tetrahedron $ABCD$ whose edges $AB$ and $CD$ have lengths $a$ and $% b$ respectively. The distance between the skew lines $AB$ and $CD$ is $d,$ and the angle between them is $\omega $. Tetrahedron $ABCD$ is divided into two solids by plane $\varepsilon $, parallel to lines $AB$ and $CD.$ The ratio of the distances of $\varepsilon $ from $AB$ and $CD$ is equal to $k.$ Compute the ratio of the volumes of the two solids obtained. \subsection{1965/4.} Find all sets of four real numbers $x_{1},x_{2},x_{3},x_{4}$ such that the sum of any one and the product of the other three is equal to $2.$ \subsection{1965/5.} Consider $\Delta OAB$ with acute angle $AOB.$ Through a point $M\neq O$ perpendiculars are drawn to $OA$ and $OB,$ the feet of which are $P$ and $Q$ respectively. The point of intersection of the altitudes of $\Delta OPQ$ is $% H.$ What is the locus of $H$ if $M$ is permitted to range over (a) the side $% AB,$ (b) the interior of $\Delta OAB$? \subsection{1965/6.} In a plane a set of $n$ points ($n\geq 3$) is given. Each pair of points is connected by a segment. Let $d$ be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length $d.$ Prove that the number of diameters of the given set is at most $% n.$ \end{document}