\hfill \thepage} %} \input{tcilatex} \begin{document} \section{Twelfth International Olympiad, 1970} \subsection{1970/1.} Let $M$ be a point on the side $AB$ of $\Delta ABC.$ Let $r_{1},r_{2}$ and $r $ be the radii of the inscribed circles of triangles $AMC,BMC$ and $ABC.$ Let $q_{1},q_{2}$ and $q$ be the radii of the escribed circles of the same triangles that lie in the angle $ACB.$ Prove that \[ \frac{r_{1}}{q_{1}}\cdot \frac{r_{2}}{q_{2}}=\frac{r}{q}. \] \subsection{1970/2.} Let $a,b$ and $n$ be integers greater than $1,$ and let $a$ and $b$ be the bases of two number systems. $A_{n-1}$ and $A_{n}$ are numbers in the system with base $a$, and $B_{n-1}$ and $B_{n}$are numbers in the system with base $% b;$ these are related as follows: \begin{eqnarray*} A_{n} &=&x_{n}x_{n-1}\cdots x_{0},A_{n-1}=x_{n-1}x_{n-2}\cdots x_{0}, \\ B_{n} &=&x_{n}x_{n-1}\cdots x_{0},B_{n-1}=x_{n-1}x_{n-2}\cdots x_{0}, \\ x_{n} &\neq &0,x_{n-1}\neq 0. \end{eqnarray*} Prove: \[ \frac{A_{n-1}}{A_{n}}<\frac{B_{n-1}}{B_{n}}\text{ if and only if }a>b. \] \subsection{1970/3.} The real numbers $a_{0},a_{1},...,a_{n},...$ satisfy the condition: \[ 1=a_{0}\leq a_{1}\leq a_{2}\leq \cdots \leq a_{n}\leq \cdots . \] The numbers $b_{1},b_{2},...,b_{n},...$ are defined by \[ b_{n}=\sum_{k=1}^{n}\left( 1-\frac{a_{k-1}}{a_{k}}\right) \frac{1}{\sqrt{% a_{k}}}. \] (a) Prove that $0\leq b_{n}<2$ for all $n.$ (b) Given $c$ with $0\leq c<2,$ prove that there exist numbers $% a_{0},a_{1},...$ with the above properties such that $b_{n}>c$ for large enough $n.$ \subsection{1970/4.} Find the set of all positive integers $n$ with the property that the set $% \{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two sets such that the product of the numbers in one set equals the product of the numbers in the other set. \subsection{1970/5.} In the tetrahedron $ABCD,$ angle $BDC$ is a right angle. Suppose that the foot $H$ of the perpendicular from $D$ to the plane $ABC$ is the intersection of the altitudes of $\Delta ABC.$ Prove that \[ (AB+BC+CA)^{2}\leq 6(AD^{2}+BD^{2}+CD^{2}). \] For what tetrahedra does equality hold? \subsection{1970/6.} In a plane there are $100$ points, no three of which are collinear. Consider all possible triangles having these points as vertices. Prove that no more than $70\%$ of these triangles are acute-angled. \end{document}