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Balkan MO Shortlist
2020 Balkan MO Shortlist
N3
N3
Part of
2020 Balkan MO Shortlist
Problems
(1)
Divisibility in function
Source: Balkan MO Shortlist 2020 N3
9/14/2021
Given an integer
k
≥
2
k\geq 2
k
≥
2
, determine all functions
f
f
f
from the positive integers into themselves such that
f
(
x
1
)
!
+
f
(
x
2
)
!
+
⋯
f
(
x
k
)
!
f(x_1)!+f(x_2)!+\cdots f(x_k)!
f
(
x
1
)!
+
f
(
x
2
)!
+
⋯
f
(
x
k
)!
is divisibe by
x
1
!
+
x
2
!
+
⋯
x
k
!
x_1!+x_2!+\cdots x_k!
x
1
!
+
x
2
!
+
⋯
x
k
!
for all positive integers
x
1
,
x
2
,
⋯
x
k
x_1,x_2,\cdots x_k
x
1
,
x
2
,
⋯
x
k
.
A
l
b
a
n
i
a
Albania
A
l
bania
function
number theory
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