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2023 LMT Spring
8
2023 Spring LMT Team #8
2023 Spring LMT Team #8
Source:
August 10, 2023
algebra
Problem Statement
Let
x
,
y
x, y
x
,
y
, and
z
z
z
be positive reals that satisfy the system
{
x
2
+
x
y
+
y
2
=
10
x
2
+
x
z
+
z
2
=
20
y
2
+
y
z
+
z
2
=
30
\begin{cases} x^2 + x y + y^2 = 10 \\ x^2 + xz + z^2 = 20 \\ y^2 + yz + z^2 = 30\end{cases}
⎩
⎨
⎧
x
2
+
x
y
+
y
2
=
10
x
2
+
x
z
+
z
2
=
20
y
2
+
yz
+
z
2
=
30
Find
x
y
+
y
z
+
x
z
x y + yz + xz
x
y
+
yz
+
x
z
.
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