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2020 BMT Fall
22
2020 BMT Team 22
2020 BMT Team 22
Source:
January 9, 2022
algebra
system of equations
Problem Statement
Suppose that
x
,
y
x, y
x
,
y
, and
z
z
z
are positive real numbers satisfying
{
x
2
+
x
y
+
y
2
=
64
y
2
+
y
z
+
z
2
=
49
z
2
+
z
x
+
x
2
=
57
\begin{cases} x^2 + xy + y^2 = 64 \\ y^2 + yz + z^2 = 49 \\ z^2 + zx + x^2 = 57 \end{cases}
⎩
⎨
⎧
x
2
+
x
y
+
y
2
=
64
y
2
+
yz
+
z
2
=
49
z
2
+
z
x
+
x
2
=
57
Then
x
y
z
3
\sqrt[3]{xyz}
3
x
yz
can be expressed as
m
/
n
m/n
m
/
n
, where
m
m
m
and
n
n
n
are relatively prime positive integers. Compute
m
+
n
m + n
m
+
n
.
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