Question

[Lê Thị Hồng Vân]

1, Cho a, b, c thỏa mãn :

\(\left\{{}\begin{matrix}\left(a+b\right)\left(b+c\right)\left(c+a\right)=abc\\\left(a^3+b^3\right)\left(b^3+c^3\right)\left(c^3+a^3\right)=a^3b^3c^3\end{matrix}\right.\\ CMR:abc=0\)

2, a, CMR nếu x + y + z = 0 thì :

\(2\left(x^5+y^5+z^5\right)=5xyz\left(x^2+y^2+z^2\right)\)

b, Cho a, b,c, d thỏa mãn : a + b + c + d = 0

CMR : \(a^3+b^3+c^3+d^3=3\left(ab-cd\right)\left(c+d\right)\)

Mọi người giải giúp mk, đc bài nào hay bài ấy nhé!

Answer 1

[Khôi Bùi]

2 ) b )

\(a+b+c+d=0\)

\(\Leftrightarrow a+b=-\left(c+d\right)\)

\(\Leftrightarrow\left(a+b\right)^3=-\left(c+d\right)^3\)

\(\Leftrightarrow a^3+b^3+3a^2b+3b^2a=-c^3-3c^2d-3d^2c-d^3\)

\(\Leftrightarrow a^3+b^3+3a^2b+3b^2a+c^3+3c^2d+3d^2c+d^3=0\)

\(\Leftrightarrow a^3+b^3+c^3+d^3=-3a^2b-3b^2a-3c^2d-3d^2c\)

\(\Leftrightarrow a^3+b^3+c^3+d^3=-3ab\left(a+b\right)-3cd\left(c+d\right)\)

\(\Leftrightarrow a^3+b^3+c^3+d^3=3ab\left(c+d\right)-3cd\left(c+d\right)\)

\(\Leftrightarrow a^3+b^3+c^3+d^3=3\left(ab-cd\right)\left(c+d\right)\) \(\left(đpcm\right)\)