cho a/x=b/y=c/z=1/5 va x+y+z khac 0 tinh A=x+y+z/a+b+c
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x-y-z=0
=> x=y+z
y=x-z
-z=y-x
B=(1-z/x)(1-x/y)(1+y/z)
B=((x-z)/x)((y-x)/y)((z+y)/z)
B=(y/x)(-z/y)(x/z)
B=(-z.y.x)/(x.y.z)
B=-1
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\(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=\frac{ayz+bxz+cxy}{xyz}=0\Rightarrow ayz+bxz+cxy=0\)
\(\left(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\right)^2=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+2\left(\frac{xy}{ab}+\frac{xz}{ac}+\frac{yz}{bc}\right)\)
\(=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+2\left(\frac{cxy+bxz+ayz}{abc}\right)=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+2\cdot\frac{0}{abc}=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\)
\(\Rightarrow\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=\left(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\right)^2\)mà \(\left(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\right)^2=1^2=1\)
\(\Rightarrow\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\)
Áp dụng tính chất dãy tỉ số bằng nhau ta có
\(\frac{a}{x}=\frac{b}{y}=\frac{c}{z}=\frac{a+b+c}{x+y+z}=\frac{1}{5}\)
\(\Rightarrow A=\frac{x+y+z}{a+b+c}=\frac{5}{1}=5\)
Vậy A = 5