Difference between revisions of "2014 UMO Problems/Problem 5"
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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. | ||
| − | < | + | <cmath> |
| − | x^2 + y^2 + z^2 & = 4x\sqrt{yz}- 2yz \end{align*}</ | + | \begin{align*} xyz & = 1\\ |
| + | x^2 + y^2 + z^2 & = 4x\sqrt{yz}- 2yz \end{align*} | ||
| + | </cmath> | ||
| + | == Solution == | ||
| + | By AM-GM | ||
| + | <cmath>x^2+y^2+z^2 + 2yz\ge x^2 + 4yz\ge 4x\sqrt{yz}</cmath> | ||
| + | Hence, the second equation implies that <math>y=z</math> and <math>x^2=4yz\implies x=2y=2z</math>. | ||
| − | = | + | Now we plug it into the first equation to get <math>(x,y,z) = \left(\sqrt[3]{4}, \frac{\sqrt[3]4}{2}, \frac{\sqrt[3]4}{2}\right)</math> |
== See Also == | == See Also == | ||
Latest revision as of 16:48, 12 February 2020
Problem
Find all positive real numbers 
, and 
that satisfy both of the following equations.
Solution
By AM-GM
![\[x^2+y^2+z^2 + 2yz\ge x^2 + 4yz\ge 4x\sqrt{yz}\]](http://latex.artofproblemsolving.com/e/8/f/e8f87922d5bf45ea825cba594c5d9d8752b1df8a.png)
Hence, the second equation implies that 
and 
.
Now we plug it into the first equation to get ![$(x,y,z) = \left(\sqrt[3]{4}, \frac{\sqrt[3]4}{2}, \frac{\sqrt[3]4}{2}\right)$](http://latex.artofproblemsolving.com/6/1/5/615f8e13b09796d4a9f54c927e71f534b91d44e4.png)
See Also
| 2014 UMO (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 1 • 2 • 3 • 4 • 5 • 6 | ||
| All UMO Problems and Solutions | ||
