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MAA AMC
USAMO
1983 USAMO
1983 USAMO
Part of
USAMO
Subcontests
(5)
3
1
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USAMO 1983 Problem 3 - Closed intervals
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.
5
1
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USAMO 1983 Problem 5 - Open interval of length 1/n
Consider an open interval of length
1
/
n
1/n
1/
n
on the real number line, where
n
n
n
is a positive integer. Prove that the number of irreducible fractions
p
/
q
p/q
p
/
q
, with
1
≤
q
≤
n
1\le q\le n
1
≤
q
≤
n
, contained in the given interval is at most
(
n
+
1
)
/
2
(n+1)/2
(
n
+
1
)
/2
.
4
1
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USAMO 1983 Problem 4 - Constructing altitude of terahedron
Six segments
S
1
,
S
2
,
S
3
,
S
4
,
S
5
,
S_1, S_2, S_3, S_4, S_5,
S
1
,
S
2
,
S
3
,
S
4
,
S
5
,
and
S
6
S_6
S
6
are given in a plane. These are congruent to the edges
A
B
,
A
C
,
A
D
,
B
C
,
B
D
,
AB, AC, AD, BC, BD,
A
B
,
A
C
,
A
D
,
BC
,
B
D
,
and
C
D
CD
C
D
, respectively, of a tetrahedron
A
B
C
D
ABCD
A
BC
D
. Show how to construct a segment congruent to the altitude of the tetrahedron from vertex
A
A
A
with straight-edge and compasses.
2
1
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USAMO 1983 Problem 2 - Roots of Quintic
Prove that the roots of
x
5
+
a
x
4
+
b
x
3
+
c
x
2
+
d
x
+
e
=
0
x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0
x
5
+
a
x
4
+
b
x
3
+
c
x
2
+
d
x
+
e
=
0
cannot all be real if
2
a
2
<
5
b
2a^2 < 5b
2
a
2
<
5
b
.
1
1
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USAMO 1983 Problem 1 - Probability of disjoint triangle
On a given circle, six points
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
,
E
E
E
, and
F
F
F
are chosen at random, independently and uniformly with respect to arc length. Determine the probability that the two triangles
A
B
C
ABC
A
BC
and
D
E
F
DEF
D
EF
are disjoint, i.e., have no common points.