Difference between revisions of "1983 USAMO Problems"
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==Problem 2== | ==Problem 2== | ||
− | Prove that the roots of<cmath>x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0</cmath> cannot all be real if <math>2a^2 < 5b</math>. | + | Prove that the roots of <cmath>x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0</cmath> cannot all be real if <math>2a^2 < 5b</math>. |
[[1983 USAMO Problems/Problem 2 | Solution]] | [[1983 USAMO Problems/Problem 2 | Solution]] |
Latest revision as of 11:25, 18 July 2016
Problems from the 1983 USAMO.
Problem 1
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.
Problem 2
Prove that the roots of cannot all be real if .
Problem 3
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.
Problem 4
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.
Problem 5
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 .
See Also
1983 USAMO (Problems • Resources) | ||
Preceded by 1982 USAMO |
Followed by 1984 USAMO | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.