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National and Regional Contests
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PEN H Problems
18
H 18
H 18
Source:
May 25, 2007
Diophantine Equations
pen
Problem Statement
Determine all positive integer solutions
(
x
,
y
,
z
,
t
)
(x, y, z, t)
(
x
,
y
,
z
,
t
)
of the equation
(
x
+
y
)
(
y
+
z
)
(
z
+
x
)
=
x
y
z
t
(x+y)(y+z)(z+x)=xyzt
(
x
+
y
)
(
y
+
z
)
(
z
+
x
)
=
x
yz
t
for which
gcd
(
x
,
y
)
=
gcd
(
y
,
z
)
=
gcd
(
z
,
x
)
=
1
\gcd(x, y)=\gcd(y, z)=\gcd(z, x)=1
g
cd
(
x
,
y
)
=
g
cd
(
y
,
z
)
=
g
cd
(
z
,
x
)
=
1
.
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