1985 OIM Problems/Problem 3
Revision as of 23:46, 8 April 2024 by Clarkculus (talk | contribs)
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, ![$\frac{r_1}{2}+\frac{r_2}{4}+\frac{r_3}{5}+\frac{r_4}{8}\ge4\sqrt[4]{r_1r_2r_3r_4\frac{1}{5(64)}}=1, so by the equality condition$](http://latex.artofproblemsolving.com/5/d/3/5d3019a969257140ddb0590d9b0a584d98c862fb.png)
\frac{r_1}{2}=\frac{r_2}{4}=\frac{r_3}{5}=\frac{r_4}{8}=\frac14
(r_1,r_2,r_3,r_4)=\boxed{(\frac12,1,\frac54,2)}$.
