Difference between revisions of "2014 UMO Problems/Problem 5"
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== Solution == | == 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 15:48, 12 February 2020
Problem
Find all positive real numbers , and that satisfy both of the following equations.
Solution
By AM-GM Hence, the second equation implies that and .
Now we plug it into the first equation to get
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 |