2018 USAJMO Problems/Problem 2
Problem
Let be positive real numbers such that . Prove that
Solution 1
WLOG let . Add to both sides of the inequality and factor to get: By substituting , we get: The last inequality is true by AM-GM. Since all these steps are reversible, the proof is complete.
Solution 2
WLOG let . Note that the equations are homogeneous, so WLOG let . Thus, the inequality now becomes , which simplifies to .
Now we will use the condition. Letting and , we have .
Plugging this into the inequality, we have , which is true since .
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
Solution 3
https://wiki-images.artofproblemsolving.com//6/69/IMG_8946.jpg
-srisainandan6
See also
2018 USAJMO (Problems • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |