Question

giúp em chị Thuy Nguyen ơi
A. cho a,b,c thỏa mãn a+b+c=0 cmr
\(2\left(a^5+b^5+c^5\right)=5abc\left(a+b+c\right)\)

B. cho x,y,a,b thỏa mãn x+y=a+b, \(x^4+y^4=a^4+b^4\)

cmr:  \(x^n+y^n=a^n+b^n\), với mọi n thuộc N*

Answer 1

Đề bài đúng phải là : Cho a,b,c thỏa mãn a+b+c=0 . CMR : \(2\left(a^5+b^5+c^5\right)=5abc\left(a^2+b^2+c^2\right)\)

a) Từ \(a+b+c=0\Rightarrow b+c=-a\Rightarrow\left(b+c\right)^5=-a^5\)

\(\Rightarrow b^5+5b^4c+10b^3c^2+10b^2c^3+5bc^4+c^5=-a^5\)

\(\Rightarrow\left(a^5+b^5+c^5\right)+5bc\left(b^3+2b^2c+2bc^2+c^3\right)=0\)

\(\Rightarrow\left(a^5+b^5+c^5\right)+5bc\left[\left(b+c\right)\left(b^2-bc+c^2\right)+2bc\left(b+c\right)\right]=0\)

\(\Rightarrow\left(a^5+b^5+c^5\right)+5bc\left(b+c\right)\left(b^2+bc+c^2\right)=0\)

\(\Rightarrow2\left(a^5+b^5+c^5\right)-5abc\left[\left(b^2+2bc+c^2\right)+b^2+c^2\right]=0\)

\(\Rightarrow2\left(a^5+b^5+c^5\right)=5abc\left[\left(b+c\right)^2+b^2+c^2\right]\)

Vậy : \(2\left(a^5+b^5+c^5\right)=5abc\left(a^2+b^2+c^2\right)\)