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

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 $(\sqrt[3]{20},-\frac{3\sqrt[3]{20}}{ 2} , \frac{5\sqrt[3]{20}}{ 2})$

2017 UNM-PNM Contest II (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10
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2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 5

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Solution

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