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2022-IMOC
A4
Mapping from the set of bijective functions
Mapping from the set of bijective functions
Source: 2022 IMOC A4
September 5, 2022
function
algebra
Problem Statement
Let the set of all bijective functions taking positive integers to positive integers be
B
.
\mathcal B.
B
.
Find all functions
F
:
B
→
R
\mathbf F:\mathcal B\to \mathbb R
F
:
B
→
R
such that
(
F
(
p
)
+
F
(
q
)
)
2
=
F
(
p
∘
p
)
+
F
(
p
∘
q
)
+
F
(
q
∘
p
)
+
F
(
q
∘
q
)
(\mathbf F(p)+\mathbf F(q))^2=\mathbf F(p \circ p)+\mathbf F(p\circ q)+\mathbf F(q\circ p)+\mathbf F(q\circ q)
(
F
(
p
)
+
F
(
q
)
)
2
=
F
(
p
∘
p
)
+
F
(
p
∘
q
)
+
F
(
q
∘
p
)
+
F
(
q
∘
q
)
for all
p
,
q
∈
B
.
p,q \in \mathcal B.
p
,
q
∈
B
.
Proposed by ckliao914
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