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\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}
