\hfill \thepage} %} \input{tcilatex} \begin{document} \section{Second International Olympiad, 1960} \subsection{1960/1.} Determine all three-digit numbers $N$ having the property that $N$ is divisible by $11,$ and $N/11$ is equal to the sum of the squares of the digits of $N.$ \subsection{1960/2.} For what values of the variable $x$ does the following inequality hold: \[ \frac{4x^{2}}{(1-\sqrt{1+2x})^{2}}<2x+9? \] \subsection{1960/3.} In a given right triangle $ABC,$ the hypotenuse $BC,$ of length $a,$ is divided into $n$ equal parts ($n$ an odd integer). Let $\alpha $ be the acute angle subtending, from $A,$ that segment which contains the midpoint of the hypotenuse. Let $h$ be the length of the altitude to the hypotenuse of the triangle. Prove: \[ \tan \alpha =\frac{4nh}{(n^{2}-1)a}. \] \subsection{1960/4.} Construct triangle $ABC,$ given $h_{a},h_{b}$ (the altitudes from $A$ and $B$% ) and $m_{a}$, the median from vertex $A.$ \subsection{1960/5.} Consider the cube $ABCDA^{\prime }B^{\prime }C^{\prime }D^{\prime }$ (with face $ABCD$ directly above face $A^{\prime }B^{\prime }C^{\prime }D^{\prime } $). (a) Find the locus of the midpoints of segments $XY,$ where $X$ is any point of $AC$ and $Y$ is any point of $B^{\prime }D^{\prime }.$ (b) Find the locus of points $Z$ which lie on the segments $XY$ of part (a) with $ZY=2XZ.$ \subsection{1960/6. } Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. Let $V_{1}$ be the volume of the cone and $V_{2}$ the volume of the cylinder. (a) Prove that $V_{1}\neq V_{2}$. (b) Find the smallest number $k$ for which $V_{1}=kV_{2}$, for this case, construct the angle subtended by a diameter of the base of the cone at the vertex of the cone. \subsection{1960/7.} An isosceles trapezoid with bases $a$ and $c$ and altitude $h$ is given. (a) On the axis of symmetry of this trapezoid, find all points $P$ such that both legs of the trapezoid subtend right angles at $P.$ (b) Calculate the distance of $P$ from either base. (c) Determine under what conditions such points $P$ actually exist. (Discuss various cases that might arise.) \end{document}