Difference between revisions of "2014 UMO Problems/Problem 5"

m (See Also)
m (Problem)
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Find all positive real numbers <math>x, y</math>, and <math>z</math> that satisfy both of the following equations.
 
Find all positive real numbers <math>x, y</math>, and <math>z</math> that satisfy both of the following equations.
<math>\begin{align*} xyz & = 1\\
+
<cmath>
x^2 + y^2 + z^2 & = 4x\sqrt{yz}- 2yz \end{align*}</math>
+
\begin{align*} xyz & = 1\\
 
+
x^2 + y^2 + z^2 & = 4x\sqrt{yz}- 2yz \end{align*}
 +
</cmath>
  
 
== Solution ==
 
== Solution ==

Revision as of 23:22, 1 February 2015

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

See Also

2014 UMO (ProblemsAnswer KeyResources)
Preceded by
Problem 4
Followed by
Problem 6
1 2 3 4 5 6
All UMO Problems and Solutions
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