\hfill \thepage} %} \input{tcilatex} \begin{document} \section{Seventeenth International Olympiad, 1975} \subsection{1975/1.} Let $x_{i},y_{i}\;(i=1,2,...,n)$ be real numbers such that \[ x_{1}\geq x_{2}\geq \cdots \geq x_{n}\text{ and }y_{1}\geq y_{2}\geq \cdots \geq y_{n}. \] Prove that, if $z_{1},z_{2},\cdots ,z_{n}$ is any permutation of $% y_{1},y_{2},\cdots ,y_{n}$, then \[ \sum_{i=1}^{n}(x_{i}-y_{i})^{2}\leq \sum_{i=1}^{n}(x_{i}-z_{i})^{2}. \] \subsection{1975/2. } Let $a_{1},a_{2},a_{3,}\cdots $ be an infinite increasing sequence of positive integers. Prove that for every $p\geq 1$ there are infinitely many $% a_{m}$ which can be written in the form \[ a_{m}=xa_{p}+ya_{q} \] with $x,y$ positive integers and $q>p.$ \subsection{1975/3. } On the sides of an arbitrary triangle $ABC,$ triangles $ABR,BCP,CAQ$ are constructed externally with $\angle CBP=\angle CAQ=45^{\circ },\angle BCP=\angle ACQ=30^{\circ },\angle ABR=\angle BAR=15^{\circ }.$ Prove that $% \angle QRP=90^{\circ }$ and $QR=RP.$ \subsection{1975/4. } When $4444^{4444}$ is written in decimal notation, the sum of its digits is $% A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $% B.$ ($A$ and $B$ are written in decimal notation.) \subsection{1975/5. } Determine, with proof, whether or not one can find $1975$ points on the circumference of a circle with unit radius such that the distance between any two of them is a rational number. \subsection{1975/6. } Find all polynomials $P,$ in two variables, with the following properties: (i) for a positive integer $n$ and all real $t,x,y$% \[ P(tx,ty)=t^{n}P(x,y) \] (that is, $P$ is homogeneous of degree $n$), (ii) for all real $a,b,c,$% \[ P(b+c,a)+P(c+a,b)+P(a+b,c)=0, \] (iii) $P(1,0)=1.$ \end{document}