1983 USAMO Problems
Problems from the 1983 USAMO.
On a given circle, six points , , , , , and are chosen at random, independently and uniformly with respect to arc length. Determine the probability that the two triangles and are disjoint, i.e., have no common points.
Prove that the roots of cannot all be real if .
Each set of a finite family of subsets of a line is a union of two closed intervals. Moreover, any three of the sets of the family have a point in common. Prove that there is a point which is common to at least half the sets of the family.
Six segments and are given in a plane. These are congruent to the edges and , respectively, of a tetrahedron . Show how to construct a segment congruent to the altitude of the tetrahedron from vertex with straight-edge and compasses.
Consider an open interval of length on the real number line, where is a positive integer. Prove that the number of irreducible fractions , with , contained in the given interval is at most .
|1983 USAMO (Problems • Resources)|
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