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9
SMT 2023 Algebra #9
SMT 2023 Algebra #9
Source:
May 3, 2023
Problem Statement
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be nonzero numbers, not necessarily real, such that
(
x
−
y
)
2
+
(
y
−
z
)
2
+
(
z
−
x
)
2
=
24
y
z
(x-y)^2+(y-z)^2+(z-x)^2=24yz
(
x
−
y
)
2
+
(
y
−
z
)
2
+
(
z
−
x
)
2
=
24
yz
and
x
2
y
z
+
y
2
z
x
+
z
2
x
y
=
3.
\tfrac{x^2}{yz}+\tfrac{y^2}{zx}+\tfrac{z^2}{xy}=3.
yz
x
2
+
z
x
y
2
+
x
y
z
2
=
3.
Compute
x
2
y
z
\tfrac{x^2}{yz}
yz
x
2
.
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