1971 IMO Problems
Problems of the 13th IMO 1971 in Czechoslovakia.
Prove that the following assertion is true for and , and that it is false for every other natural number :
If are arbitrary real numbers, then
Consider a convex polyhedron with nine vertices ; let be the polyhedron obtained from by a translation that moves vertex to . Prove that at least two of the polyhedra have an interior point in common.
Prove that the set of integers of the form contains an infinite subset in which every two members are relatively prime.
All the faces of tetrahedron are acute-angled triangles. We consider all closed polygonal paths of the form defined as follows: is a point on edge distinct from and ; similarly, are interior points of edges , respectively. Prove:
(a) If , then among the polygonal paths, there is none of minimal length.
(b) If , then there are infinitely many shortest polygonal paths, their common length being , where .
Prove that for every natural number , there exists a finite set of points in a plane with the following property: For every point in , there are exactly points in which are at unit distance from .
Let be a square matrix whose elements are non-negative integers. Suppose that whenever an element , the sum of the elements in the th row and the th column is . Prove that the sum of all the elements of the matrix is .