Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2018 UMO Problems/Problem 2"

(→‎Solution 1)
m (→‎Solution 1)
Line 8: Line 8:
  
 
Plugging in <math>x = 0</math>, we find that <math>abc = 1</math>. Using AM-GM, we have that <math>a+b+c \leq 3 ^3\sqrt{abc} = \fbox{3}</math>
 
Plugging in <math>x = 0</math>, we find that <math>abc = 1</math>. Using AM-GM, we have that <math>a+b+c \leq 3 ^3\sqrt{abc} = \fbox{3}</math>
 +
~bigbrain123

Revision as of 16:09, 31 July 2021

Problem 2

Let $P(x)$ be a cubic polynomial $(x-a)(x-b)(x-c)$, where $a, b,$ and $c$ are positive real numbers. Let Q(x) be the polynomial with $Q(x) = (x-ab)(x-bc)(x-ac)$. If $P(x) = Q(x)$ for all $x$, then find the minimum possible value of $a + b + c$.

Solution 1

Plugging in $x = 0$, we find that $abc = 1$. Using AM-GM, we have that $a+b+c \leq 3 ^3\sqrt{abc} = \fbox{3}$ ~bigbrain123

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2018_UMO_Problems/Problem_2&oldid=159342"