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National and Regional Contests
Iran Contests
Iran Team Selection Test
2023 Iran Team Selection Test
5
function from Z to R
function from Z to R
Source: Iran TST 2023 ; Exam 1 Problem 5
March 15, 2023
function
algebra
Problem Statement
Suppose that
n
≥
2
n\ge2
n
≥
2
and
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
are natural numbers that
(
a
1
,
a
2
,
.
.
.
,
a
n
)
=
1
(a_1,a_2,...,a_n)=1
(
a
1
,
a
2
,
...
,
a
n
)
=
1
. Find all strictly increasing function
f
:
Z
→
R
f: \mathbb{Z} \to \mathbb{R}
f
:
Z
→
R
that:
∀
x
1
,
x
2
,
.
.
.
,
x
n
∈
Z
:
f
(
∑
i
=
1
n
x
i
a
i
)
=
∑
i
=
1
n
f
(
x
i
a
i
)
\forall x_1,x_2,...,x_n \in \mathbb{Z} : f(\sum_{i=1}^{n} {x_ia_i}) = \sum_{i=1}^{n} {f(x_ia_i})
∀
x
1
,
x
2
,
...
,
x
n
∈
Z
:
f
(
i
=
1
∑
n
x
i
a
i
)
=
i
=
1
∑
n
f
(
x
i
a
i
)
Proposed by Navid Safaei and Ali Mirzaei
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