1983 USAMO Problems

Problems from the 1982 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

1983 USAMO (Problems • Resources)
Preceded by 1982 USAMO	Followed by 1984 USAMO
1 • 2 • 3 • 4 • 5
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1983 USAMO Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

See Also