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\hfill \thepage} %} \input{tcilatex} \begin{document} \section{Tenth International Olympiad, 1968} \subsection{1968/1.} Prove that there is one and only one triangle whose side lengths are consecutive integers, and one of whose angles is twice as large as another. \subsection{1968/2.} Find all natural numbers $x$ such that the product of their digits (in decimal notation) is equal to $x^{2}-10x-22.$ \subsection{1968/3.} Consider the system of equations \begin{eqnarray*} ax_{1}^{2}+bx_{1}+c &=&x_{2} \\ ax_{2}^{2}+bx_{2}+c &=&x_{3} \\ &&\cdots \\ ax_{n-1}^{2}+bx_{n-1}+c &=&x_{n} \\ ax_{n}^{2}+bx_{n}+c &=&x_{1}, \end{eqnarray*} with unknowns $x_{1},x_{2},\cdots ,x_{n}$, where $a,b,c$ are real and $a\neq 0.$ Let $\Delta =(b-1)^{2}-4ac.$ Prove that for this system (a) if $\Delta <0,$ there is no solution, (b) if $\Delta =0,$ there is exactly one solution, (c) if $\Delta >0,$ there is more than one solution. \subsection{1968/4.} Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which are the sides of a triangle. \subsection{1968/5.} Let $f$ be a real-valued function defined for all real numbers $x$ such that, for some positive constant $a,$ the equation $f(x+a)=\frac{1}{2}+\sqrt{f(x)-[f(x)]^{2}}$ holds for all $x.$ (a) Prove that the function $f$ is periodic (i.e., there exists a positive number $b$ such that $f(x+b)=f(x)$ for all $x$). (b) For $a=1,$ give an example of a non-constant function with the required properties. \subsection{1968/6.} For every natural number $n,$ evaluate the sum $\sum_{k=0}^{\infty }\left[ \frac{n+2^{k}}{2^{k+1}}\right] =\left[ \frac{n+1}{% 2}\right] +\left[ \frac{n+2}{4}\right] +\cdots +\left[ \frac{n+2^{k}}{2^{k+1}% }\right] +\cdots$ (The symbol $[x]$ denotes the greatest integer not exceeding $x.$) \end{document}