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Turkey Team Selection Test
1997 Turkey Team Selection Test
2
Turkey TST 1997 Problem 2, how many pairs
Turkey TST 1997 Problem 2, how many pairs
Source: Turkey TST 1997 Problem 2
November 30, 2011
algebra proposed
algebra
Problem Statement
The sequences
(
a
n
)
(a_{n})
(
a
n
)
,
(
b
n
)
(b_{n})
(
b
n
)
are defined by
a
1
=
α
a_{1} = \alpha
a
1
=
α
,
b
1
=
β
b_{1} = \beta
b
1
=
β
,
a
n
+
1
=
α
a
n
−
β
b
n
a_{n+1} = \alpha a_{n} - \beta b_{n}
a
n
+
1
=
α
a
n
−
β
b
n
,
b
n
+
1
=
β
a
n
+
α
b
n
b_{n+1} = \beta a_{n} + \alpha b_{n}
b
n
+
1
=
β
a
n
+
α
b
n
for all
n
>
0.
n > 0.
n
>
0.
How many pairs
(
α
,
β
)
(\alpha, \beta)
(
α
,
β
)
of real numbers are there such that
a
1997
=
b
1
a_{1997} = b_{1}
a
1997
=
b
1
and
b
1997
=
a
1
b_{1997} = a_{1}
b
1997
=
a
1
?
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