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National and Regional Contests
North Macedonia Contests
Macedonian Team Selection Test
2021 Macedonian Team Selection Test
Problem 5
Problem 5
Part of
2021 Macedonian Team Selection Test
Problems
(1)
Functional Inequality with Divisibility Condition
Source: 2021 Macedonian Team Selection Test P5
5/30/2021
Determine all functions
f
:
N
→
N
f:\mathbb{N}\to \mathbb{N}
f
:
N
→
N
such that for all
a
,
b
∈
N
a, b \in \mathbb{N}
a
,
b
∈
N
the following conditions hold:
(
i
)
(i)
(
i
)
f
(
f
(
a
)
+
b
)
∣
b
a
−
1
f(f(a)+b) \mid b^a-1
f
(
f
(
a
)
+
b
)
∣
b
a
−
1
;
(
i
i
)
(ii)
(
ii
)
f
(
f
(
a
)
)
≥
f
(
a
)
−
1
f(f(a))\geq f(a)-1
f
(
f
(
a
))
≥
f
(
a
)
−
1
.
Divisibility
functions
algebra