2023 SSMO Speed Round Problems/Problem 9
Problem
Find the sum of the maximum and minimum values of 
under the constraint that 
Solution
We want to maximize 
such
![\[8x^2+7xy+5y^2 = \frac{k}{88}(3x^2+5xy+3y^2)\]](http://latex.artofproblemsolving.com/9/6/9/9696a5a32800bf4cab9e3f0887ef81202890fe7d.png)
or, if 
![\[8a^2+7a+5 = \frac{k}{88}(3a^2+5a+3)\]](http://latex.artofproblemsolving.com/2/2/2/2223aa66daadfad0f8338f01b5715b416b0d3c80.png)
which has discriminant in 
of
![\[121(-k^2 + 688k - 78144)\]](http://latex.artofproblemsolving.com/3/a/0/3a041093c2f1d46c4371fab7134cdb8b9fdfb2b9.png)
so the sum of the extremes of are 
.
