Problem
ALG-B2-M12-P018 Set 5. Squares versus cubes
#18
★★★★★ Level 5 of 5
Prove for \(a,b,c\ge0\): \[(a^2+b^2+c^2)^3\ge(a^3+b^3+c^3)^2.\]
Hint. Use Cauchy for \(\sum a\cdot a^2\).
By Cauchy, \((\sum a^3)^2\le(\sum a^2)(\sum a^4)\). Also, \(\sum a^4\le(\sum a^2)^2\). Hence \((\sum a^3)^2\le(\sum a^2)^3\).
Mock set 5: the problem is intended for independent method selection without an explicit cue in the statement.