cho x/y+z + y/z+x +z/x+y =1. Tính S=x^2/y+z +y^2/z+x + z^2/x+y

Question

cho x/y+z + y/z+x +z/x+y =1. Tính S=x^2/y+z +y^2/z+x + z^2/x+y

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1 tháng 2021-11-08T20:45:42+00:00 2 Answers 3 views 0

1. Ta có:

$\dfrac{x}{y+z}+\dfrac{y}{z+x}+\dfrac{z}{x+y}=1$

$\to (x+y+z)(\dfrac{x}{y+z}+\dfrac{y}{z+x}+\dfrac{z}{x+y})=1.(x+y+z)$

$\to (x+y+x)(\dfrac{x}{y+z}+\dfrac{y}{z+x}+\dfrac{z}{x+y})=x+y+z$(*)

Mà ta có:

$(x+y+z)(\dfrac{x}{y+z}+\dfrac{x^2}{y+z}+\dfrac{xy}{y+z}+\dfrac{xz}{y+z}$

$\to = \dfrac{x^2}{y+z}+\dfrac{x(y+z)}{y+z}$(**)

$\to = \dfrac{x^2}{y+z}+x$

Tương tự:

$(x+y+z)\dfrac{y}{z+x}=\dfrac{y^2}{z+x}+y$(***)

$(x+y+x)\dfrac{x}{x+y}=\dfrac{z^2}{x+y}+z$(****)

Từ (*),(**),(***),(****),suy ra:

$\dfrac{x^2}{y+z}+x+\dfrac{y^2}{z+x}+y+\dfrac{z^2}{x+y}=x+y+z$

$\to \dfrac{x^2}{y+z}+\dfrac{y^2}{z+x}+\dfrac{z^2}{x+y}=0$

Mà:

$S=\dfrac{x^2}{y+z}+\dfrac{y^2}{x+z}+\dfrac{z^2}{x+y}$

$\to S=0$

2. Ta có: x/(y+z)+y/(x+z)+z/(x+y)=1

⇒(x+y+z)(x/(y+z)+y/(x+z)+z/(x+y))=x+y+z

⇒(x^2+x(y+z))/(y+z)+(y^2+y(x+z))/(x+z)+(z^2+z(x+y))/(x+y)=x+y+z

⇒x^2/(y+z)+x+y^2/(z+x)+y+z^2/(x+y)+z=x+y+z

⇒x^2/(y+z)+y^2/(z+x)+z^2/(x+y)=0

Vậy S=0