MathDB
Problems
Contests
National and Regional Contests
Taiwan Contests
TST Round 2
2021 Taiwan TST Round 2
5
5
Part of
2021 Taiwan TST Round 2
Problems
(1)
Easy Functional Equation
Source: 2021 Taiwan TST Round 2 Mock Day 2 P5
4/10/2021
Let
∥
x
∥
∗
=
(
∣
x
∣
+
∣
x
−
1
∣
−
1
)
/
2
\|x\|_*=(|x|+|x-1|-1)/2
∥
x
∥
∗
=
(
∣
x
∣
+
∣
x
−
1∣
−
1
)
/2
. Find all
f
:
N
→
N
f:\mathbb{N}\to\mathbb{N}
f
:
N
→
N
such that f^{(\|f(x)-x\|_*)}(x)=x, \forall x\in\mathbb{N}. Here
f
(
0
)
(
x
)
=
x
f^{(0)}(x)=x
f
(
0
)
(
x
)
=
x
and
f
(
n
)
(
x
)
=
f
(
f
(
n
−
1
)
(
x
)
)
f^{(n)}(x)=f(f^{(n-1)}(x))
f
(
n
)
(
x
)
=
f
(
f
(
n
−
1
)
(
x
))
for all
n
∈
N
n\in\mathbb{N}
n
∈
N
.Proposed by usjl
algebra
functional equation
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