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7
2022 PUMaC Team #7
2022 PUMaC Team #7
Source:
September 9, 2023
algebra
Problem Statement
Pick
x
,
y
,
z
x, y, z
x
,
y
,
z
to be real numbers satisfying
(
−
x
+
y
+
z
)
2
−
1
3
=
4
(
y
−
z
)
2
(-x+y+z)^2-\frac13 = 4(y-z)^2
(
−
x
+
y
+
z
)
2
−
3
1
=
4
(
y
−
z
)
2
,
(
x
−
y
+
z
)
2
−
1
4
=
4
(
z
−
x
)
2
(x-y+z)^2-\frac14 = 4(z-x)2
(
x
−
y
+
z
)
2
−
4
1
=
4
(
z
−
x
)
2
, and
(
x
+
y
−
z
)
2
−
1
5
=
4
(
x
−
y
)
2
(x+y-z)^2 -\frac15 = 4(x-y)^2
(
x
+
y
−
z
)
2
−
5
1
=
4
(
x
−
y
)
2
. If the value of
x
y
+
y
z
+
z
x
xy+yz +zx
x
y
+
yz
+
z
x
can be written as
p
q
\frac{p}{q}
q
p
for relatively prime positive integers
p
,
q
p, q
p
,
q
, find
p
+
q
p + q
p
+
q
.
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