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Kürschák Math Competition
2002 Kurschak Competition
2
2
Part of
2002 Kurschak Competition
Problems
(1)
Where Fibonacci meets Farey
Source: Kürschák 2002, problem 2
7/8/2014
The Fibonacci sequence is defined as
f
1
=
f
2
=
1
f_1=f_2=1
f
1
=
f
2
=
1
,
f
n
+
2
=
f
n
+
1
+
f
n
f_{n+2}=f_{n+1}+f_n
f
n
+
2
=
f
n
+
1
+
f
n
(
n
∈
N
n\in\mathbb{N}
n
∈
N
). Suppose that
a
a
a
and
b
b
b
are positive integers such that
a
b
\frac ab
b
a
lies between the two fractions
f
n
f
n
−
1
\frac{f_n}{f_{n-1}}
f
n
−
1
f
n
and
f
n
+
1
f
n
\frac{f_{n+1}}{f_{n}}
f
n
f
n
+
1
. Show that
b
≥
f
n
+
1
b\ge f_{n+1}
b
≥
f
n
+
1
.
number theory unsolved
number theory