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China Western Mathematical Olympiad
2024 China Western Mathematical Olympiad
7
Divisible problem
Divisible problem
Source: 2024 CWMO P7
August 7, 2024
number theory
Problem Statement
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be four positive integers such that
a
>
b
>
c
>
d
a>b>c>d
a
>
b
>
c
>
d
. Given that
a
b
+
b
c
+
c
a
+
d
2
∣
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
ab+bc+ca+d^2|(a+b)(b+c)(c+a)
ab
+
b
c
+
c
a
+
d
2
∣
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
. Find the minimal value of
Ω
(
a
b
+
b
c
+
c
a
+
d
2
)
\Omega (ab+bc+ca+d^2)
Ω
(
ab
+
b
c
+
c
a
+
d
2
)
. Here
Ω
(
n
)
\Omega(n)
Ω
(
n
)
denotes the number of prime factors
n
n
n
has. e.g.
Ω
(
12
)
=
3
\Omega(12)=3
Ω
(
12
)
=
3
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