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1997 AIME Problems
14
Roots of Unity
Roots of Unity
Source:
December 20, 2005
Problem Statement
Let
v
v
v
and
w
w
w
be distinct, randomly chosen roots of the equation
z
1997
−
1
=
0.
z^{1997}-1=0.
z
1997
−
1
=
0.
Let
m
/
n
m/n
m
/
n
be the probability that
2
+
3
≤
∣
v
+
w
∣
,
\sqrt{2+\sqrt{3}}\le |v+w|,
2
+
3
≤
∣
v
+
w
∣
,
where
m
m
m
and
n
n
n
are relatively prime positive integers. Find
m
+
n
.
m+n.
m
+
n
.
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