International Mathematical Olympiad · 1971

Problems

Problem 1Prove that the following assertion is true for $n=3$ and $n=5$ , and false for every other natural number $n>2$ : for arbitrary real numbers $a_1,a_2,\ldots,a_n$ , $(a_1-a_2)(a_1-a_3)\cdots(a_1-a_n)+(a_2-a_1)(a_2-a_3)\cdots(a_2-a_n)+\cdots+(a_n-a_1)(a_n-a_2)\cdots(a_n-a_{n-1})\ge0$ .Solutions: 1
Problem 2Consider a convex polyhedron $P_1$ with nine vertices $A_1,A_2,\ldots,A_9$ . Let $P_i$ be obtained from $P_1$ by a translation that moves $A_1$ to $A_i$ ( $i=2,3,\ldots,9$ ). Prove that at least two of $P_1,P_2,\ldots,P_9$ have an interior point in common.Solutions: 1
Problem 3Prove 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.Solutions: 1
Problem 4All faces of tetrahedron $ABCD$ are acute-angled triangles. Consider closed polygonal paths $XYZTX$ , where $X$ is interior to $AB$ , and $Y,Z,T$ are interior to $BC,CD,DA$ , respectively. Prove: (a) if $\angle DAB+\angle BCD\ne\angle CDA+\angle ABC$ , no path has minimal length; (b) if $\angle DAB+\angle BCD=\angle CDA+\angle ABC$ , infinitely many shortest paths exist, with common length $2AC\sin(\alpha/2)$ , where $\alpha=\angle BAC+\angle CAD+\angle DAB$ .Solutions: 1
Problem 5Prove that for every natural number $m$ , there exists a finite set $S$ of points in a plane such that every point $A$ in $S$ has exactly $m$ points in $S$ at unit distance from $A$ .Solutions: 1
Problem 6Let $A=(a_{ij})$ ( $i,j=1,2,\ldots,n$ ) be a square matrix whose elements are nonnegative integers. Suppose that whenever $a_{ij}=0$ , 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 elements of the matrix is at least $n^2/2$ .Solutions: 1