a)\(a^2+b^2+c^2\ge ab+bc+ca\)
AM-GM:\(a^2+b^2\ge2ab\)
\(b^2+c^2\ge2bc\)
\(c^2+a^2\ge2ca\)
Cộng vế theo vế\(\Rightarrow2\left(a^2+b^2+c^2\right)\ge2\left(ab+bc+ca\right)\)
\(\Rightarrow a^2+b^2+c^2\ge ab+bc+ca\)
b)AM-GM:\(a^4+a^4+b^4+c^4\ge4a^2bc\)
\(b^4+b^4+a^4+c^4\ge4ab^2c\)
\(a^4+b^4+c^4+c^4\ge4abc^2\)
Cộng vế theo vế\(\Rightarrow4\left(a^4+b^4+c^4\right)\ge4a^2bc+4ab^2c+4abc^2\)
\(\Rightarrow a^4+b^4+c^4\ge abc\left(a+b+c\right)\left(đpcm\right)\)
c)
ta có
(a+b-c)(a-b+c) = a2-(b-c)2 ≤ a2
(a-b+c)(b+c-a) = c2-(a-b)2 ≤ b2
(a+b-c)(b+c-a) = b2-(a-c)2 ≤ b2
nhân các vế với nhau ta đc
[(a+b-c)(a-b+c)(b+c-a)]2 ≤ (abc)2
<=> (a+b-c)(a-b+c)(b+c-a) ≤ abc (đpcm)
