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Sweden Contests
Swedish Mathematical Competition
1965 Swedish Mathematical Competition
4
4
Part of
1965 Swedish Mathematical Competition
Problems
(1)
f(1/(1+2x))/f(x) indendent of x if f(x) = \frac{1 + Ax}{1 + Bx}
Source: 1965 Swedish Mathematical Competition p4
3/21/2021
Find constants
A
>
B
A > B
A
>
B
such that
f
(
1
1
+
2
x
)
f
(
x
)
\frac{f\left( \frac{1}{1+2x}\right) }{f(x)}
f
(
x
)
f
(
1
+
2
x
1
)
is independent of
x
x
x
, where
f
(
x
)
=
1
+
A
x
1
+
B
x
f(x) = \frac{1 + Ax}{1 + Bx}
f
(
x
)
=
1
+
B
x
1
+
A
x
for all real
x
≠
−
1
B
x \ne - \frac{1}{B}
x
=
−
B
1
. Put
a
0
=
1
a_0 = 1
a
0
=
1
,
a
n
+
1
=
1
1
+
2
a
n
a_{n+1} = \frac{1}{1 + 2a_n}
a
n
+
1
=
1
+
2
a
n
1
. Find an expression for an by considering
f
(
a
0
)
,
f
(
a
1
)
,
.
.
.
f(a_0), f(a_1), ...
f
(
a
0
)
,
f
(
a
1
)
,
...
.
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