MathDB
Problems
Contests
National and Regional Contests
Taiwan Contests
TST Round 1
2017 Taiwan TST Round 1
3
3
Part of
2017 Taiwan TST Round 1
Problems
(1)
Functional equation 006
Source: 2017 Taiwan TST Round 1, Day 4, Problem 3
4/14/2017
Find all injective functions
f
:
N
→
N
f:\mathbb{N} \to \mathbb{N}
f
:
N
→
N
such that
f
f
(
a
)
(
b
)
f
f
(
b
)
(
a
)
=
(
f
(
a
+
b
)
)
2
f^{f\left(a\right)}\left(b\right)f^{f\left(b\right)}\left(a\right)=\left(f\left(a+b\right)\right)^2
f
f
(
a
)
(
b
)
f
f
(
b
)
(
a
)
=
(
f
(
a
+
b
)
)
2
holds for all
a
,
b
∈
N
a,b \in \mathbb{N}
a
,
b
∈
N
. Note that
f
k
(
n
)
f^{k}\left(n\right)
f
k
(
n
)
means
f
(
f
(
…
f
⏟
k
(
n
)
…
)
)
\underbrace{f(f(\ldots f}_{k}(n) \ldots ))
k
f
(
f
(
…
f
(
n
)
…
))
algebra
functional equation