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\noindent {\bf 17th IMO 1975}
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\noindent {\bf A1}. Let $x_1\ge x_2\ge\ldots\ge x_n$, and $y_1\ge y_2\ge\ldots\ge y_n$ be real numbers. Prove that if $z_i$ is any permutation of the $y_i$, then: $$\sum_1^n(x_i-y_i)^2\le\sum_1^n(x_i-z_i)^2.$$
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\noindent {\bf A2}. Let $a_1<a_2<a_3<\ldots$ be positive integers. Prove that for every $i\ge1$, there are infinitely many $a_n$ that can be written in the form $a_n=ra_i+sa_j$, with $r,s$ positive integers and $j>i$.
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\noindent {\bf A3}. Given any triangle $ABC$, construct external triangles $ABR,BCP,CAQ$ on the sides, so that $\angle PBC=45^o,\angle PCB=30^o,\angle QAC=45^o,\angle QCA=30^o,\angle RAB=15^o,\angle RBA=15^o$. Prove that $\angle QRP=90^o$ and $QR=RP$.
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\noindent {\bf B1}. Let $A$ be the sum of the decimal digits of $4444^{4444}$, and $B$ the sum of the decimal digits of $A$. Find the sum of the decimal digits of $B$.
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\noindent {\bf B2}. Find $1975$ points on the circumference of a unit circle such that the distance between each pair is rational, or prove it impossible.
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\noindent {\bf B3}. Find all polynomials $P(x,y)$ in two variables such that:

(1) $P(tx,ty)=t^nP(x,y)$ for some positive integer $n$ and all real $t,x,y$:

(2) for all real $x,y,z: P(y+z,x)+P(z+x,y)+P(x+y,z)=0$;

(3) $P(1,0)=1$.
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\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 19 August 2003

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