Question

如何证明a^4+b^4+c^4≧a^2b^2+b^2c^2+c^2a^2≥(a+b+c)abc?

Answer 1

a^4+b^4+c^4=2(a^4+b^4+c^4)/2=

[(a^4+b^4)+(b^4+c^4)+(a^4+c^4)]/2

≥(2a^2b^2+2b^2c^2+2c^2a^2)/2≥a^2b^2+b^2c^2+a^2c^2;a^2b^2+b^2c^2+c^2a^2=[a^2(b^2+c^2)+c^2(a^2+b^2)+b^2(a^2+c^2)]/2≥a^2bc+c^2ab+b^2ac≥abc(a+c+b).