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Benelux
2024 Benelux
4
4
Part of
2024 Benelux
Problems
(1)
2024 BxMO P4
Source: 2024 BxMO P4
4/28/2024
For each positive integer
n
n
n
, let
r
a
d
(
n
)
rad(n)
r
a
d
(
n
)
denote the product of the distinct prime factors of
n
n
n
. Show that there exists integers
a
,
b
>
1
a,b > 1
a
,
b
>
1
such that
g
c
d
(
a
,
b
)
=
1
gcd(a,b)=1
g
c
d
(
a
,
b
)
=
1
and
r
a
d
(
a
b
(
a
+
b
)
)
<
a
+
b
202
4
2024
rad(ab(a+b)) < \frac{a+b}{2024^{2024}}
r
a
d
(
ab
(
a
+
b
))
<
202
4
2024
a
+
b
.For example,
r
a
d
(
20
)
=
r
a
d
(
2
2
⋅
5
)
=
2
⋅
5
=
10
rad(20)=rad(2^2\cdot 5)=2\cdot 5=10
r
a
d
(
20
)
=
r
a
d
(
2
2
⋅
5
)
=
2
⋅
5
=
10
.
number theory