Question

Cho a + b + c = 0 và \(a^2+b^2+c^2=1\) . CMR \(a^4+b^4+c^4=\frac{1}{2}\)

Answer 1

\(\left(a+b+c\right)^2=0\)

\(\Rightarrow a^2+b^2+c^2+2\left(ab+bc+ac\right)=0\)

\(\Rightarrow ab+bc+ac=-\frac{1}{2}\)

\(\Rightarrow a^2b^2+b^2c^2+a^2c^2+2abc\left(a+b+c\right)=\frac{1}{4}\)

\(\Rightarrow a^2b^2+b^2c^2+a^2c^2=\frac{1}{4}\)

\(\left(a^2+b^2+c^2\right)^2=1\)

\(\Rightarrow a^4+b^4+c^4+2\left(a^2b^2+b^2c^2+a^2c^2\right)=1\)

\(\Rightarrow a^4+b^4+c^4+\frac{1}{2}=1\)

\(\Rightarrow...\)