For all ordered triples $(p,q,r)$ define the polynomial
$$f_{p,q,r}(x)=x^3-px^2+qx-r$$
Let $a_{1},a_{2},a_{3},b_{1},b_{2},b_{3},c_{1},c_{2},c_{3}$ be (not necessarily distinct) positive reals such that the roots of $f_{a_{1},a_{2},a_{3}}(x)$ are $b_{1},b_{2},b_{3}$ and the roots of $f_{b_{1},b_{2},b_{3}}(x)$ are $c_{1},c_{2},c_{3}$. Determine the maximum possible value of
$$ \frac{9\sqrt[3]{b_{3}}}{b_{1}+3} + \frac{4+3b_{1}+2b_{2}+b_{3}}{a_{1}+1} $$
I used Vieta's formulas combined with calculus. I set this expression equal to $y$ and then cubed both sides. Then I tried to use the fact that since $y$ is real, the cubic in $y$ (generated by cubing both sides) will have three real roots. Now, I differentiated the equation w.r.t. $y$ (assuming everything else to be constant). I got a quadratic in $y$ and I then made its discriminant $>0$. Now I used Vieta's formulas. After that I'm stuck since I still have more than one variable left. Also, I'm not yet familiar with multivariable calculus. Any help will be greately appreciated.
Thanks!
Best Answer
This question is an old problem posed on Brilliant.
You can view it here, along with the solution by the problem creator Zi Song.