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Polish MO Finals
1975 Polish MO Finals
6
6
Part of
1975 Polish MO Finals
Problems
(1)
f(1/2)=1975/ 2^{1975}
Source: Polish MO Finals 1975 p6
8/23/2024
On the interval
[
0
,
1
]
[0,1]
[
0
,
1
]
are given functions
S
(
x
)
=
1
−
x
S(x) = 1 - x
S
(
x
)
=
1
−
x
and
T
(
x
)
=
x
/
2
T(x) = x/2
T
(
x
)
=
x
/2
. Does there exist a function of the form
f
=
g
1
∘
g
2
∘
.
.
.
∘
g
n
f = g_1\circ g_2\circ ... \circ g_n
f
=
g
1
∘
g
2
∘
...
∘
g
n
, where
n
∈
N
n \in N
n
∈
N
and each
g
k
g_k
g
k
is either
S
(
x
)
S(x)
S
(
x
)
or
T
(
x
)
T(x)
T
(
x
)
, such that
f
(
1
2
)
=
1975
2
1975
?
f\left(\frac12\right)=\frac{1975}{2^{1975}} \, ?
f
(
2
1
)
=
2
1975
1975
?
algebra