Ta có:

\(\frac{3}{x+y}=\frac{2}{y+z}=\frac{1}{z+x}\Rightarrow\frac{x+y}{3}=\frac{y+z}{2}=\frac{z+x}{1}=\frac{x+y+y+z+z+x}{3+2+1}=\frac{2\left(x+y+z\right)}{6}=\frac{x+y+z}{3}\)

\(\frac{x+y+z}{3}=\frac{x+y}{3}\Rightarrow z=0\)

Thay vào P, ta có:

\(P=\frac{2x+2y+2019z}{x+y-2020z}=\frac{2x+2y}{x+y}=\frac{2\left(x+y\right)}{x+y}=2\)

Vậy P=2

giúp mình với nhé!

Ta có:

\(\frac{xy}{x+y}=\frac{yz}{y+z}=\frac{zx}{z+x}\rightarrow\frac{x+y}{xy}=\frac{y+z}{yz}=\frac{z+x}{zx}\)

\(\Rightarrow\frac{1}{x}+\frac{1}{y}=\frac{1}{y}+\frac{1}{z}=\frac{1}{z}+\frac{1}{x}\Rightarrow\frac{1}{x}=\frac{1}{y}=\frac{1}{z}\Rightarrow x=y=z\)

Thay tất cả giá trị x,y,z vào M ta được:

\(M=\frac{2020x^3+2020y^3+2020z^3}{x^3+y^3+z^3}+\frac{2021x^5+2021y^5}{x^5+y^5}\)

\(\Rightarrow M=\frac{2020\left(x^3+y^3+z^3\right)}{x^3+y^3+z^3}+\frac{2021\left(x^5+y^5\right)}{x^5+y^5}\)

\(\Rightarrow M=2020+2021=4041\)

\(\frac{2x+2y-z}{z}=\frac{2x-y+2z}{y}=\frac{-x+2y+2z}{x} \)

=>\(\frac{2x+2y-z}{z}+3=\frac{2x-y+2z}{y}+3=\frac{-x+2y+2z}{x}+3\)

=>\(\frac{2x+2y+2z}{z}=\frac{2x+2y+2z}{y}=\frac{2x+2y+2z}{x}\)

=>\(\frac{x+y+z}{z}=\frac{x+y+z}{y}=\frac{x+y+z}{x}\)

=>\(\orbr{\begin{cases}x+y+z=0\\x=y=z\end{cases}}\)

Với \(x+y+z=0\Rightarrow\hept{\begin{cases}x+y=-z\\y+z=-x\\x+z=-y\end{cases}}\)

\(\Rightarrow M=\frac{\left(x+y\right)\left(y+z\right)\left(x+z\right)}{8xyz}=\frac{-xyz}{8xyz}=-\frac{1}{8}\)

Với \(x=y=z\)\(\Rightarrow M=\frac{\left(x+y\right)\left(y+z\right)\left(x+z\right)}{8xyz}=\frac{2x.2y.2z}{8xyz}=\frac{8xyz}{8xyz}=1\)

Áp dụng t/c của dãy tỉ số bằng nhau, ta có:

 \(\frac{x-2y+z}{y}=\frac{z-2x+y}{x}=\frac{x-2z+y}{z}=\frac{x-2y+z+z-2x+y+x-2z+y}{x+y+z}=0\)(vì x;y;z \(\ne\)0)

=> \(\hept{\begin{cases}\frac{x-2y+z}{y}=0\\\frac{z-2x+y}{x}=0\\\frac{x-2z+y}{z}=0\end{cases}}\) => \(\hept{\begin{cases}x-2y+z=0\\z-2x+y=0\\x-2z+y=0\end{cases}}\) => \(\hept{\begin{cases}x+z=2y\\y+z=2x\\x+y=2z\end{cases}}\) 

Khi đó, ta có: A = \(\left(1+\frac{y}{x}\right)\left(1+\frac{z}{y}\right)\left(1+\frac{x}{z}\right)+2020\)

=> A = \(\left(\frac{x+y}{x}\right)\left(\frac{y+z}{y}\right)\left(\frac{x+z}{z}\right)+2020\)

=> A = \(\frac{2z}{x}\cdot\frac{2x}{y}\cdot\frac{2y}{z}+2020\)

=> A = \(8+2020=2028\)