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Turkey EGMO TST
2024 Turkey EGMO TST
2
2
Part of
2024 Turkey EGMO TST
Problems
(1)
Euler totient and functional equation
Source: 2024 Turkey EGMO TST P2
2/12/2024
Find all functions
f
:
Z
+
→
Z
+
f:\mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}
f
:
Z
+
→
Z
+
such that the conditions a) a-b \mid f(a)-f(b) for all
a
≠
b
a\neq b
a
=
b
and
a
,
b
∈
Z
+
a,b \in \mathbb{Z}^{+}
a
,
b
∈
Z
+
b) f(\varphi(a))=\varphi(f(a)) for all
a
∈
Z
+
a \in \mathbb{Z}^{+}
a
∈
Z
+
where
φ
\varphi
φ
is the Euler's totient function. holds
number theory
function
Divisibility
functional equation