Problem
ALG-B2-M06-P014 Fourth powers versus abc
#14
★★★★☆ Level 4 of 5
Prove for \(a,b,c\ge0\): \[a^4+b^4+c^4\ge abc(a+b+c).\]
Hint. This symmetric inequality is linear in \(r\) for fixed \(p,q\). Check \(b=c\) and the boundary.
By UVW, it is enough to check the two-equal case and the boundary. If \(b=c=1\), \(a=t\), the difference is \((t-1)^2(t^2+2t+2)\ge0\). If one variable is \(0\), the right side is \(0\), while the left side is nonnegative. Hence the inequality holds.
The first full application of UVW reduction.