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National and Regional Contests
North Macedonia Contests
Macedonian Team Selection Test
2022 Macedonian Team Selection Test
Problem 3
Problem 3
Part of
2022 Macedonian Team Selection Test
Problems
(1)
integer valued functions with at most two values for f(2022)
Source: Macedonian TST 2022, P3
5/21/2022
We consider all functions
f
:
N
→
N
f: \mathbb{N} \rightarrow \mathbb{N}
f
:
N
→
N
such that
f
(
f
(
n
)
+
n
)
=
n
f(f(n)+n)=n
f
(
f
(
n
)
+
n
)
=
n
and
f
(
a
+
b
−
1
)
≤
f
(
a
)
+
f
(
b
)
f(a+b-1) \leq f(a)+f(b)
f
(
a
+
b
−
1
)
≤
f
(
a
)
+
f
(
b
)
for all positive integers
a
,
b
,
n
a, b, n
a
,
b
,
n
. Prove that there are at most two values for
f
(
2022
)
f(2022)
f
(
2022
)
.
Proposed
by
Ilija
Jovcheski
\textit {Proposed by Ilija Jovcheski}
Proposed by Ilija Jovcheski
algebra
function
functional equation