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2022 Taiwan Mathematics Olympiad
1
1
Part of
2022 Taiwan Mathematics Olympiad
Problems
(1)
Coprime integers with product being perfect squares
Source: 2022 Taiwan Mathematics Olympiad
2/19/2022
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be three positive integers with
gcd
(
x
,
y
,
z
)
=
1
\gcd(x,y,z)=1
g
cd
(
x
,
y
,
z
)
=
1
. If
x
∣
y
z
(
x
+
y
+
z
)
,
x\mid yz(x+y+z),
x
∣
yz
(
x
+
y
+
z
)
,
y
∣
x
z
(
x
+
y
+
z
)
,
y\mid xz(x+y+z),
y
∣
x
z
(
x
+
y
+
z
)
,
z
∣
x
y
(
x
+
y
+
z
)
,
z\mid xy(x+y+z),
z
∣
x
y
(
x
+
y
+
z
)
,
and
x
+
y
+
z
∣
x
y
z
,
x+y+z\mid xyz,
x
+
y
+
z
∣
x
yz
,
show that
x
y
z
(
x
+
y
+
z
)
xyz(x+y+z)
x
yz
(
x
+
y
+
z
)
is a perfect square.Proposed by usjl
number theory
Taiwan