\hfill \thepage} %} \input{tcilatex} \begin{document} \section{Ninth International Olympiad, 1967} \subsection{1967/1.} Let $ABCD$ be a parallelogram with side lengths $AB=a,AD=1,$ and with $% \angle BAD=\alpha $. If $\Delta ABD$ is acute, prove that the four circles of radius $1$ with centers $A,B,C,D$ cover the parallelogram if and only if \[ a\leq \cos \alpha +\sqrt{3}\sin \alpha . \] \subsection{1967/2.} Prove that if one and only one edge of a tetrahedron is greater than $1,$ then its volume is $\leq 1/8.$ \subsection{1967/3.} Let $k,m,n$ be natural numbers such that $m+k+1$ is a prime greater than $% n+1.$ Let $c_{s}=s(s+1).$ Prove that the product \[ (c_{m+1}-c_{k})(c_{m+2}-c_{k})\cdots (c_{m+n}-c_{k}) \] is divisible by the product $c_{1}c_{2}\cdots c_{n}$. \subsection{1967/4.} Let $A_{0}B_{0}C_{0}$ and $A_{1}B_{1}C_{1}$ be any two acute-angled triangles. Consider all triangles $ABC$ that are similar to $\Delta A_{1}B_{1}C_{1}$ (so that vertices $A_{1},B_{1},C_{1}$ correspond to vertices $A,B,C,$ respectively) and circumscribed about triangle $% A_{0}B_{0}C_{0}$ (where $A_{0}$ lies on $BC,B_{0}$ on $CA,$ and $AC_{0}$ on $% AB$). Of all such possible triangles, determine the one with maximum area, and construct it. \subsection{1967/5.} Consider the sequence $\{c_{n}\}$, where \begin{eqnarray*} c_{1} &=&a_{1}+a_{2}+\cdots +a_{8} \\ c_{2} &=&a_{1}^{2}+a_{2}^{2}+\cdots +a_{8}^{2} \\ &&\cdots \\ c_{n} &=&a_{1}^{n}+a_{2}^{n}+\cdots +a_{8}^{n} \\ &&\cdots \end{eqnarray*} in which $a_{1},a_{2},\cdots ,a_{8}$ are real numbers not all equal to zero. Suppose that an infinite number of terms of the sequence $\{c_{n}\}$ are equal to zero. Find all natural numbers $n$ for which $c_{n}=0.$ \subsection{1967/6.} In a sports contest, there were $m$ medals awarded on $n$ successive days ($% n>1$). On the first day, one medal and $1/7$ of the remaining $m-1$ medals were awarded. On the second day, two medals and $1/7$ of the now remaining medals were awarded; and so on. On the $n$-th and last day, the remaining $n$ medals were awarded. How many days did the contest last, and how many medals were awarded altogether? \end{document}