Question

Cho a,b,c thỏa mãn (a - b)² + (b - c)² + (c - a)² = 6abc

CMR: a³ + b³ + c³ = 3abc(a + b + c +1)

Answer 1

Ta có:

\(\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=6abc\)

\(\Leftrightarrow a^2+b^2+c^2-\left(ab+bc+ac\right)=3abc\)

\(\Leftrightarrow\left(a+b+c\right)^2-3\left(ab+bc+ac\right)=3abc\)

Đặt \(\left(a+b+c,ab+bc+ac,abc\right)=\left(p,q,r\right)\)

\(\Rightarrow p^2-3q=3r\)

Khi đó \(a^3+b^3+c^3=\left(a+b+c\right)^3-3\left(a+b\right)\left(b+c\right)\left(c+a\right)\)

\(\Leftrightarrow a^3+b^3+c^3=\left(a+b+c\right)^3-3\left(a+b+c\right)\left(ab+bc+ac\right)+3abc\)

\(\Leftrightarrow a^3+b^3+c^3=p^3+3pq+3r=p\left(p^2-3q\right)+3r=3pr+3r\)

Vậy .....

Chúc bạn học tốt!