IMO 1971

Problem 1. Prove that the following assertion is true for $n=3$ and $n=5$, and that it is false for every other natural number $n>2$:

If $a_1,a_2,...,a_n$ are arbitrary real numbers, then

$(a_1-a_2)(a_1-a_3)\cdots (a_1-a_n)+(a_2-a_1)(a_2-a_3)\cdots (a_2-a_n)$

$+\cdots +(a_n-a_1)(a_n-a_2)\cdots (a_n-a_{n-1})\ge 0$

Problem 2. Consider a convex polyhedron $P_1$ with nine vertices $A_1{A}_2,...,A_9;$ let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves vertex $A_1$ to $A_i(i=2,3,...,9)$. Prove that at least two of the polyhedra $P_1,P_2,...,P_9$ have an interior point in common.

Problem 3. Prove that the set of integers of the form $2^k-3(k=2,3,...)$ contains an infinite subset in which every two members are relatively prime.

Problem 4. All the faces of tetrahedron $ABCD$ are acute-angled triangles. We consider all closed polygonal paths of the form $XYZTX$ defined as follows: $X$ is a point on edge $AB$ distinct from $A$ and $B$; similarly, $Y,Z,T$ are interior points of edges $BCCD,DA$, respectively. Prove:

(a) If $\angle DAB+\angle BCD\ne \angle CDA+\angle ABC$, then among the polygonal paths, there is none of minimal length.

(b) If $\angle DAB+\angle BCD=\angle CDA+\angle ABC$, then there are infinitely many shortest polygonal paths, their common length being $2AC\sin (\alpha /2),$ where $\alpha =\angle BAC+\angle CAD+\angle DAB$.

Problem 5. Prove that for every natural number $m$, there exists a finite set $S$ of points in a plane with the following property: For every point $A$ in $S$, there are exactly $m$ points in $S$ which are at unit distance from $A$.

Problem 6. Let $A=(a_{ij})(i,j=1,2,...,n)$ be a square matrix whose elements are non-negative integers. Suppose that whenever an element $a_{ij}=0$, the sum of the elements in the $i$th row and the $j$th column is $\ge n$. Prove that the sum of all the elements of the matrix is $\ge n^2/2$.