International Competitions 1971

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 $a_1, a_2, \ldots, a_n$ are arbitrary real numbers, then $$(a_1 - a_2)(a_1 - a_3) \cdots (a_1 - a_n) + (a_2 - a_1)(a_2 - a_3) \cdots (a_2 - a_n)$$ $$+ \cdots + (a_n - a_1)(a_n - a_2) \cdots (a_n - a_{n-1}) \geq 0$$

Consider a convex polyhedron $P_1$ with nine vertices $A_1A_2, \ldots, A_9$; let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves vertex $A_1$ to $A_i$ $(i = 2, 3, \ldots, 9)$. Prove that at least two of the polyhedra $P_1, P_2, \ldots, P_9$ have an interior point in common.

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

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 $BC$, $CD$, $DA$, respectively. Prove:

(a) If $\angle DAB + \angle BCD \neq \angle CDA + \angle ABC$, 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 $\alpha = \angle BAC + \angle CAD + \angle DAB$.

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 $A$.

Let $A = (a_{ij})$ $(i, j = 1, 2, \ldots, n)$ 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 $\geq n$. Prove that the sum of all the elements of the matrix is $\geq n^2/2$.