Cho (a + b + c)2 = a2 + b2 + c2 và a, b, c khác 0.

* Hướng dẫn giải

Ta có: (a + b + c)² = a² + b² + c²

$\iff$ ab + bc + ca = 0

Mà a, b, c khác 0 nên $\frac{1}{a}$ + $\frac{1}{b}$ + $\frac{1}{c}$= 0

$$\begin{gathered} \iff \frac{1}{a}+\frac{1}{b}=-\frac{1}{c} \\ \iff {\left(\frac{1}{a}+\frac{1}{b}\right)}^3={\left(-\frac{1}{c}\right)}^3 \\ \iff \frac{1}{a^3}+\frac{1}{b^3}+3.\frac{1}{a}.\frac{1}{b}.\left(\frac{1}{a}+\frac{1}{b}\right)=-\frac{1}{c^3} \end{gathered}$$

$\iff \frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}=\frac{3}{abc}$(đpcm).