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\noindent {\bf 13th IMO 1971}
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\noindent {\bf A1}. Let $E_n=(a_1-a_2)(a_1-a_3)\ldots(a_1-a_n)+(a_2-a_1)(a_2-a_3)\ldots(a_2-a_n)+\ldots+(a_n-a_1)(a_n-a_2)\ldots(a_n-a_{n-1})$. Let $S_n$ be the proposition that $E_n\ge0$ for all real $a_i$. Prove that $S_n$ is true for $n=3$ and $5$, but for no other $n>2$.
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\noindent {\bf A2}. Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.
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\noindent {\bf A3}. Prove that we can find an infinite set of positive integers of the from $2^n-3$ (where $n$ is a positive integer) every pair of which are relatively prime.
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\noindent {\bf B1}. All faces of the tetrahedron $ABCD$ are acute-angled. Take a point $X$ in the interior of the segment $AB$, and similarly $Y$ in $BC, Z$ in $CD$ and $T$ in $AD$.

(a) If $\angle DAB+\angle BCD\ne\angle CDA+\angle ABC$, then prove none of the closed paths $XYZTX$ has minimal length;

(b) If $\angle DAB+\angle BCD=\angle CDA+\angle ABC$, then there are infinitely many shortest paths $XYZTX$, each with length $2AC\sin k$, where $2k=\angle BAC+\angle CAD+\angle DAB$.
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\noindent {\bf B2}. Prove that for every positive integer $m$ we can find a finite set $S$ of points in the plane, such that given any point $A$ of $S$, there are exactly $m$ points in $S$ at unit distance from $A$.
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\noindent {\bf B3}. Let $A=(a_{ij})$, where $i,j=1,2,\ldots,n$, be a square matrix with all $a_{ij}$ non-negative integers. For each $i,j$ such that $a_{ij}=0$, the sum of the elements in the $i$th row and the $j$th column is at least $n$. Prove that the sum of all the elements in the matrix is at least $n^2\over2$.
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\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 19 August 2003

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