Difference between revisions of "1983 USAMO Problems"

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Problems from the '''1983 [[USAMO]].'''
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==Problem 1==
 
==Problem 1==
 
On a given circle, six points <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>, <math>E</math>, and <math>F</math> are chosen at random, independently and uniformly with respect to arc length. Determine the probability that the two triangles <math>ABC</math> and <math>DEF</math> are disjoint, i.e., have no common points.
 
On a given circle, six points <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>, <math>E</math>, and <math>F</math> are chosen at random, independently and uniformly with respect to arc length. Determine the probability that the two triangles <math>ABC</math> and <math>DEF</math> are disjoint, i.e., have no common points.
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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>.
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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 12:25, 18 July 2016

Problems from the 1983 USAMO.

Problem 1

On a given circle, six points $A$, $B$, $C$, $D$, $E$, and $F$ are chosen at random, independently and uniformly with respect to arc length. Determine the probability that the two triangles $ABC$ and $DEF$ are disjoint, i.e., have no common points.

Solution

Problem 2

Prove that the roots of \[x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0\] cannot all be real if $2a^2 < 5b$.

Solution

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.

Solution

Problem 4

Six segments $S_1, S_2, S_3, S_4, S_5,$ and $S_6$ are given in a plane. These are congruent to the edges $AB, AC, AD, BC, BD,$ and $CD$, respectively, of a tetrahedron $ABCD$. Show how to construct a segment congruent to the altitude of the tetrahedron from vertex $A$ with straight-edge and compasses.

Solution

Problem 5

Consider an open interval of length $1/n$ on the real number line, where $n$ is a positive integer. Prove that the number of irreducible fractions $p/q$, with $1\le q\le n$, contained in the given interval is at most $(n+1)/2$.

Solution

See Also

1983 USAMO (ProblemsResources)
Preceded by
1982 USAMO
Followed by
1984 USAMO
1 2 3 4 5
All USAMO Problems and Solutions

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