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2013 BMT Spring
7
2013 BMT Team 7
2013 BMT Team 7
Source:
January 5, 2022
number theory
Problem Statement
Consider the infinite polynomial
G
(
x
)
=
F
1
x
+
F
2
x
2
+
F
3
x
3
+
.
.
.
G(x) = F_1x+F_2x^2 +F_3x^3 +...
G
(
x
)
=
F
1
x
+
F
2
x
2
+
F
3
x
3
+
...
defined for
0
<
x
<
5
−
1
2
0 < x <\frac{\sqrt5 -1}{2}
0
<
x
<
2
5
−
1
where Fk is the
k
k
k
th term of the Fibonacci sequence defined to be
F
k
=
F
k
−
1
+
F
k
−
2
F_k = F_{k-1} + F_{k-2}
F
k
=
F
k
−
1
+
F
k
−
2
with
F
1
=
1
F_1 = 1
F
1
=
1
,
F
2
=
1
F_2 = 1
F
2
=
1
. Determine the value a such that
G
(
a
)
=
2
G(a) = 2
G
(
a
)
=
2
.
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