\hfill \thepage} %} \input{tcilatex} \begin{document} \section{Fourteenth International Olympiad, 1972} \subsection{1972/1.} Prove that from a set of ten distinct two-digit numbers (in the decimal system), it is possible to select two disjoint subsets whose members have the same sum. \subsection{1972/2.} Prove that if $n\geq 4,$ every quadrilateral that can be inscribed in a circle can be dissected into $n$ quadrilaterals each of which is inscribable in a circle. \subsection{1972/3.} Let $m$ and $n$ be arbitrary non-negative integers. Prove that \[ \frac{(2m)!(2n)!}{m\prime n!(m+n)!} \] is an integer. $(0!=1.)$ \subsection{1972/4.} Find all solutions $(x_{1},x_{2},x_{3},x_{4},x_{5})$ of the system of inequalities \begin{eqnarray*} (x_{1}^{2}-x_{3}x_{5})(x_{2}^{2}-x_{3}x_{5}) &\leq &0 \\ (x_{2}^{2}-x_{4}x_{1})(x_{3}^{2}-x_{4}x_{1}) &\leq &0 \\ (x_{3}^{2}-x_{5}x_{2})(x_{4}^{2}-x_{5}x_{2}) &\leq &0 \\ (x_{4}^{2}-x_{1}x_{3})(x_{5}^{2}-x_{1}x_{3}) &\leq &0 \\ (x_{5}^{2}-x_{2}x_{4})(x_{1}^{2}-x_{2}x_{4}) &\leq &0 \end{eqnarray*} where $x_{1},x_{2},x_{3},x_{4},x_{5}$ are positive real numbers. \subsection{1972/5.} Let $f$ and $g$ be real-valued functions defined for all real values of $x$ and $y,$ and satisfying the equation \[ f(x+y)+f(x-y)=2f(x)g(y) \] for all $x,y.$ Prove that if $f(x)$ is not identically zero, and if $\left| f(x)\right| $ $\leq 1$ for all $x,$ then $\left| g(y)\right| \leq 1$ for all $y.$ \subsection{1972/6.} Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane. \end{document}