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International Contests
IMO Longlists
1990 IMO Longlists
67
67
Part of
1990 IMO Longlists
Problems
(1)
Find the range of r - ILL 1990 PRK1
Source:
9/18/2010
Let
a
+
b
i
a + bi
a
+
bi
and
c
+
d
i
c + di
c
+
d
i
be two roots of the equation
x
n
=
1990
x^n = 1990
x
n
=
1990
, where
n
≥
3
n \geq 3
n
≥
3
is an integer and
a
,
b
,
c
,
d
∈
R
a,b,c,d \in \mathbb R
a
,
b
,
c
,
d
∈
R
. Under the linear transformation
f
=
(
a
c
b
d
)
f =\left(\begin{array}{cc}a&c\\b &d\end{array}\right)
f
=
(
a
b
c
d
)
, we have
(
2
,
1
)
→
(
1
,
2
)
(2, 1) \to (1, 2)
(
2
,
1
)
→
(
1
,
2
)
. Denote
r
r
r
to be the distance from the image of
(
2
,
2
)
(2, 2)
(
2
,
2
)
to the origin. Find the range of
r
.
r.
r
.
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