Difference between revisions of "2017 JBMO Problems/Problem 2"
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\end{align*}</cmath> | \end{align*}</cmath> | ||
Hence <cmath>(x+y+z)(xy+yz+xz-2)\geq 9(z)(z+1)(z+2)</cmath> | Hence <cmath>(x+y+z)(xy+yz+xz-2)\geq 9(z)(z+1)(z+2)</cmath> | ||
| + | |||
| + | == Solution 2 == | ||
| + | Expanding the equation gives | ||
| + | <cmath>x^2y + x^2z + y^2x+ y^2z + z^2x + z^2y</cmath> | ||
== See also == | == See also == | ||
Revision as of 11:07, 22 July 2021
Contents
Problem
Let 
be positive integers such that 
.Prove that ![\[(x+y+z)(xy+yz+zx-2)\geq 9xyz.\]](http://latex.artofproblemsolving.com/5/6/3/56366cbef963b44b24f2b97c3db0fc34d36dffab.png)
When does the equality hold?
Solution
Since the equation is symmetric and are distinct integers WLOG we can assume that 
.
Hence ![\[(x+y+z)(xy+yz+xz-2)\geq 9(z)(z+1)(z+2)\]](http://latex.artofproblemsolving.com/0/6/e/06e19b24b54e55eb1debd2ab894fcfd2e2e078d0.png)
Solution 2
Expanding the equation gives
![\[x^2y + x^2z + y^2x+ y^2z + z^2x + z^2y\]](http://latex.artofproblemsolving.com/d/7/d/d7d33152ab9be1da6a058b49b730d636baf7d594.png)
See also
| 2017 JBMO (Problems • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 1 • 2 • 3 • 4 | ||
| All JBMO Problems and Solutions | ||
