IMO 1971

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 $n>2$ .

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.

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.

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 $2k=\angle BAC+\angle CAD+\angle DAB$ .

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 $A$ .

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 $\frac {n^2}{2}$ .