Question

Cho x,y,z là các số thực thỏa mãn điều kiện: (x+y+z)³-x³-y³-z³=0.

Chứng minh rằng: (x¹¹+y¹¹)(y⁷+z⁷)(z²⁰¹⁷+x²⁰¹⁷)=0

Answer 1

\(\left(x+y+z\right)^3-x^3-y^3-z^3=0\)

\(\Leftrightarrow x^3+y^3+z^3+3\left(x+y\right)\left(y+z\right)\left(x+z\right)-x^3-y^3-z^3=0\)

=>3(x+y)(y+z)(x+z)=0

=>(x+y)(y+z)(x+z)=0

\(\left(x^{11}+y^{11}\right)\left(y^7+z^7\right)\left(x^{2017}+z^{2017}\right)\)

\(=\left(x+y\right)\cdot A\cdot\left(y+z\right)\cdot B\cdot\left(x+z\right)\cdot C\)

=0