2014 UMO Problems/Problem 5

Problem

Find all positive real numbers $x, y$ , and $z$ that satisfy both of the following equations. $\begin{align*} xyz & = 1\\ x^2 + y^2 + z^2 & = 4x\sqrt{yz}- 2yz \end{align*}$

Solution

By AM-GM $x^2+y^2+z^2 + 2yz\ge x^2 + 4yz\ge 4x\sqrt{yz}$ Hence, the second equation implies that $y=z$ and $x^2=4yz\implies x=2y=2z$ .

Now we plug it into the first equation to get $(x,y,z) = \left(\sqrt[3]{4}, \frac{\sqrt[3]4}{2}, \frac{\sqrt[3]4}{2}\right)$

2014 UMO (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
1 • 2 • 3 • 4 • 5 • 6
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2014 UMO Problems/Problem 5

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