2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 5

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Problem

Find all real triples $(x,y,z)$ which are solutions to the system:

$x^3 + x^2y + x^2z = 40$

$y^3 + y^2x + y^2z = 90$

$z^3 + z^2x + z^2y = 250$

Solution

There are $3$ solutions $(2,3,5)$ or $(-\sqrt[3]{\frac{40}{3}} ,\sqrt[3]{45},\sqrt[3]{\frac{5^4}{3}})$ or $(\sqrt[3]{20},-\frac{3\sqrt[3]{20}}{ 2} , \frac{5\sqrt[3]{20}}{ 2})$

See also

2017 UNM-PNM Contest II (ProblemsAnswer KeyResources)
Preceded by
Problem 4
Followed by
Problem 6
1 2 3 4 5 6 7 8 9 10
All UNM-PNM Problems and Solutions
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