\hfill \thepage} %} \input{tcilatex} \begin{document} \section{Fourth International Olympiad, 1962} \subsection{1962/1.} Find the smallest natural number $n$ which has the following properties: (a) Its decimal representation has $6$ as the last digit. (b) If the last digit $6$ is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number $% n.$ \subsection{1962/2.} Determine all real numbers $x$ which satisfy the inequality: \[ \sqrt{3-x}-\sqrt{x+1}>\frac{1}{2}. \] \subsection{1962/3.} Consider the cube $ABCDA^{\prime }B^{\prime }C^{\prime }D^{\prime }$ ($ABCD$ and $A^{\prime }B^{\prime }C^{\prime }D^{\prime }$ are the upper and lower bases, respectively, and edges $AA^{\prime },BB^{\prime },CC^{\prime },DD^{\prime }$ are parallel). The point $X$ moves at constant speed along the perimeter of the square $ABCD$ in the direction $ABCDA,$ and the point $Y $ moves at the same rate along the perimeter of the square $B^{\prime }C^{\prime }CB$ in the direction $B^{\prime }C^{\prime }CBB^{\prime }$. Points $X$ and $Y$ begin their motion at the same instant from the starting positions $A$ and $B^{\prime }$, respectively. Determine and draw the locus of the midpoints of the segments $XY.$ \subsection{1962/4.} Solve the equation $\cos ^{2}x+\cos ^{2}2x+\cos ^{2}3x=1.$ \subsection{1962/5.} On the circle $K$ there are given three distinct points $A,B,C.$ Construct (using only straightedge and compasses) a fourth point $D$ on $K$ such that a circle can be inscribed in the quadrilateral thus obtained. \subsection{1962/6.} Consider an isosceles triangle. Let $r$ be the radius of its circumscribed circle and $\rho $ the radius of its inscribed circle. Prove that the distance d between the centers of these two circles is \[ d=\sqrt{r(r-2\rho )}. \] \subsection{1962/7.} The tetrahedron $SABC$ has the following property: there exist five spheres, each tangent to the edges $SA,SB,SC,BCCA,AB,$ or to their extensions. (a) Prove that the tetrahedron $SABC$ is regular. (b) Prove conversely that for every regular tetrahedron five such spheres exist. \end{document}