1985 OIM Problems/Problem 3
Problem
Find the roots 
, 
, 
, and 
of the equation:
![\[4x^4-ax^3+bx^2-cx+5=0\]](http://latex.artofproblemsolving.com/d/9/3/d93275ae96fdc4e2a22e121aab01b3fc03049a4e.png)
knowing that they're all real, positives and that:
![\[\frac{r_1}{2}+\frac{r_2}{4}+\frac{r_3}{5}+\frac{r_4}{8}=1\]](http://latex.artofproblemsolving.com/d/1/9/d195dc3007c6d7345fb00b1bdacae607e880987d.png)
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
By Vieta's, 
. Because the roots are real and positive, by AM-GM, ![$\frac{r_1}{2}+\frac{r_2}{4}+\frac{r_3}{5}+\frac{r_4}{8}\ge4\sqrt[4]{r_1r_2r_3r_4\frac{1}{5(64)}}=1$](http://latex.artofproblemsolving.com/f/a/9/fa9096f74534c51538c0231f5ccedfb060fa668e.png)
, so by the equality condition 
, so 
.
