\hfill \thepage} %} \input{tcilatex} \begin{document} \section{Eleventh International Olympiad, 1969} \subsection{1969/1.} Prove that there are infinitely many natural numbers $a$ with the following property: the number $z=n^{4}+a$ is not prime for any natura1 number $n.$ \subsection{1969/2.} Let $a_{1},a_{2},\cdots ,a_{n}$ be real constants, $x$ a real variable, and \begin{eqnarray*} f(x) &=&\cos (a_{1}+x)+\frac{1}{2}\cos (a_{2}+x)+\frac{1}{4}\cos (a_{3}+x) \\ &&+\cdots +\frac{1}{2^{n-1}}\cos (a_{n}+x). \end{eqnarray*} Given that $f(x_{1})=f(x_{2})=0,$ prove that $x_{2}-x_{1}=m\pi $ for some integer $m.$ \subsection{1969/3.} For each value of $k=1,2,3,4,5,$ find necessary and sufficient conditions on the number $a>0$ so that there exists a tetrahedron with $k$ edges of length $a,$ and the remaining $6-k$ edges of length $1$. \subsection{1969/4.} A semicircular arc $\gamma $ is drawn on $AB$ as diameter. $C$ is a point on $\gamma $ other than $A$ and $B,$ and $D$ is the foot of the perpendicular from $C$ to $AB.$ We consider three circles, $\gamma _{1},\gamma _{2},\gamma _{3}$, all tangent to the line $AB.$ Of these, $\gamma _{1}$ is inscribed in $\Delta ABC,$ while $\gamma _{2}$ and $\gamma _{3}$ are both tangent to $CD$ and to $\gamma $, one on each side of $CD.$ Prove that $\gamma _{1},\gamma _{2}$ and $\gamma _{3}$ have a second tangent in common. \subsection{1969/5.} \vspace{1pt}Given $n>4$ points in the plane such that no three are collinear. Prove that there are at least $\binom{n-3}{2}$ convex quadrilaterals whose vertices are four of the given points. \subsection{1969/6.} Prove that for all real numbers $x_{1},x_{2},y_{1},y_{2},z_{1},z_{2},$ with $% x_{1}>0,$ $x_{2}>0,x_{1}y_{1}-z_{1}^{2}>0,x_{2}y_{2}-z_{2}^{2}>0,$ the inequality \[ \frac{8}{\left( x_{1}+x_{2}\right) \left( y_{1}+y_{2}\right) -\left( z_{1}+z_{2}\right) ^{2}}\leq \frac{1}{x_{1}y_{1}-z_{1}^{2}}+\frac{1}{% x_{2}y_{2}-z_{2}^{2}} \] is satisfied. Give necessary and sufficient conditions for equality. \end{document}