Difference between revisions of "2018 USAJMO Problems/Problem 2"
Nukelauncher (talk | contribs) (Created page with "== Problem == Let <math>a,b,c</math> be positive real numbers such that <math>a+b+c=4\sqrt[3]{abc}</math>. Prove that <cmath>2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.</cm...") |
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Revision as of 00:32, 20 April 2018
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:
The last inequality is true by AM-GM. Since all these steps are reversible, the proof is complete.
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
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 |