\hfill \thepage} %} \input{tcilatex} \begin{document} \section{First International Olympiad, 1959} \subsection{1959/1.} Prove that the fraction $\frac{21n+4}{14n+3}$ is irreducible for every natural number $n.$ \subsection{1959/2.} For what real values of $x$ is \[ \sqrt{(x+\sqrt{2x-1})}+\sqrt{(x-\sqrt{2x-1})}=A, \] given (a) $A=\sqrt{2},$ (b) $A=1$, (c) $A=2,$ where only non-negative real numbers are admitted for square roots? \subsection{1959/3.} Let $a,b,c$ be real numbers. Consider the quadratic equation in $\cos x:$ \[ a\cos ^{2}x+b\cos x+c=0. \] Using the numbers $a,b,c,$ form a quadratic equation in $\cos 2x$, whose roots are the same as those of the original equation. Compare the equations in $\cos x$ and $\cos 2x$ for $a=4,b=2,c=-1.$ \subsection{1959/4.} Construct a right triangle with given hypotenuse $c$ such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle. \subsection{1959/5.} An arbitrary point $M$ is selected in the interior of the segment $AB.$ The squares $AMCD$ and $MBEF$ are constructed on the same side of $AB,$ with the segments $AM$ and $MB$ as their respective bases. The circles circumscribed about these squares, with centers $P$ and $Q,$ intersect at $M$ and also at another point $N.$ Let $N^{\prime }$ denote the point of intersection of the straight lines $AF$ and $BC.$ (a) Prove that the points $N$ and $N^{\prime }$ coincide. (b) Prove that the straight lines $MN$ pass through a fixed point $S$ independent of the choice of $M.$ (c) Find the locus of the midpoints of the segments $PQ$ as $M$ varies between $A$ and $B.$ \subsection{1959/6.} Two planes, $P$ and $Q,$ intersect along the line $p.$ The point $A$ is given in the plane $P,$ and the point $C$ in the plane $Q;$ neither of these points lies on the straight line $p.$ Construct an isosceles trapezoid $ABCD$ (with $AB$ parallel to $CD$) in which a circle can be inscribed, and with vertices $B$ and $D$ lying in the planes $P$ and $Q$ respectively. \end{document}