Problem
ALG-B2-M12-P017 Set 5. Fourth powers
#17
★★★★☆ Level 4 of 5
Prove for \(a,b,c\ge0\): \[a^4+b^4+c^4\ge abc(a+b+c).\]
Hint. Estimate \(a^2bc\), \(ab^2c\), and \(abc^2\) separately.
By AM-GM, \(2a^4+b^4+c^4\ge4a^2bc\), and similarly for the other two mixed terms. Adding the three estimates gives \(4\sum a^4\ge4abc(a+b+c)\).
Mock set 5: the problem is intended for independent method selection without an explicit cue in the statement.