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National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1964 Dutch Mathematical Olympiad
4
g {g(x) } = g(x)
g {g(x) } = g(x)
Source: Netherlands - Dutch NMO 1964 p4
January 31, 2023
algebra
function
functional equation
Problem Statement
The function
ƒ
ƒ
ƒ
is defined at
[
0
,
1
]
[0,1]
[
0
,
1
]
, and
f
{
f
(
x
)
}
=
ƒ
(
x
)
f\{f(x)\} = ƒ(x)
f
{
f
(
x
)}
=
ƒ
(
x
)
.
∃
c
∈
[
0
,
1
]
[
f
(
c
)
=
1
2
]
\exists _{c\in [0,1]} \left[f(c) =\frac12 \right]
∃
c
∈
[
0
,
1
]
[
f
(
c
)
=
2
1
]
Determine
f
(
1
2
)
.
f\left(\frac12 \right).
f
(
2
1
)
.
∀
t
∈
[
0
,
1
]
∃
s
∈
[
0
,
1
]
[
f
(
s
)
=
t
]
\forall _{t\in [0,1]}\exists _{s\in [0,1]}[f(s) = t]
∀
t
∈
[
0
,
1
]
∃
s
∈
[
0
,
1
]
[
f
(
s
)
=
t
]
. Determine
f
f
f
.Prove that the function
g
g
g
, with
g
(
x
)
=
x
g(x) = x
g
(
x
)
=
x
,
0
≤
x
≤
k
0 \le x \le k
0
≤
x
≤
k
,
g
(
x
)
=
k
g(x) = k
g
(
x
)
=
k
,
k
≤
x
≤
1
k \le x \le 1
k
≤
x
≤
1
satisfies the relation
g
{
g
(
x
)
}
=
g
(
x
)
g\{g(x)\} = g(x)
g
{
g
(
x
)}
=
g
(
x
)
.
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