cdn (storage)
by OronSH, Aug 27, 2024, 2:07 AM
Brandon Wang wrote:
Prove that ![\[ \sum_{\text{sym}} \left( a^8b^4 + a^7b^4c
- a^8b^2c^2 - a^6b^6 \right) \ge 0 \]](//latex.artofproblemsolving.com/f/e/9/fe9884a9b78119b9b459e68869d916d7da6dd0fc.png)
holds for all 
.
![\[ \sum_{\text{sym}} \left( a^8b^4 + a^7b^4c
- a^8b^2c^2 - a^6b^6 \right) \ge 0 \]](http://latex.artofproblemsolving.com/f/e/9/fe9884a9b78119b9b459e68869d916d7da6dd0fc.png)
holds for all 
.![\[(a^2-b^2)^2(b^2-c^2)^2(c^2-a^2)^2+abc\left(\sum_{\text{cyc}}c^3(a-b)^2\left(\frac14(a^2+b^2)^2+\frac34(a^2-b^2)^2\right)\right)\]](http://latex.artofproblemsolving.com/c/3/c/c3c5034fe57f756ac0706a63b504f1310d3252c9.png)
which is clearly nonnegative.which is clearly nonnegative.
now we want to try and make it simpler, such as by removing some layers. the outer layer is a 
hexagon ring, and it reminds us of the inequality 
which in cdn is
and is very sharp which is good. thus we want to subtract this, but its too small so we can "inflate it" (this is possible because its just the same as replacing 
with 
)
now this looks simpler and that is because its just three copies of
summed together along different directions. now this is just a degree 
polynomial after dehomogenizing so we can try to factor it. and it factors as 
which is positive (the second term is a weighted average of two squares). done4 Comments
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This post has been edited 1 time. Last edited by OronSH, Aug 27, 2024, 2:18 AM
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pikapika007, Aug 27, 2024, 2:09 AM
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