Cho x + y + z = 2007 và \(\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{x+z}=\frac{1}{90}\). Tính tổng : \(\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}\).
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8 tháng 1 2018
Ta có : \(\left(x+y+z\right)\left(\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{z+x}\right)=\) = \(\frac{2017}{672}\)
\(\Leftrightarrow\frac{x+y+z}{x+y}+\frac{x+y+z}{y+z}+\frac{x+y+z}{z+x}=\)\(\frac{2017}{672}\)
\(\Leftrightarrow1+\frac{z}{x+y}+1+\frac{x}{y+z}+1+\frac{z}{z+x}\)= \(\frac{2017}{672}\)
\(\Rightarrow A=\frac{2017}{672}-3\)
GH
19 tháng 6 2023
\(\dfrac{y+z-x}{x}=\dfrac{z+x-y}{y}=\dfrac{x+y-z}{z}\\ \Rightarrow\dfrac{y+z-x}{x}+2=\dfrac{z+x-y}{y}+2=\dfrac{x+y-z}{z}+2\\ \Rightarrow\dfrac{x+y+z}{x}=\dfrac{x+y+z}{y}=\dfrac{x+y+z}{z}\\ \Rightarrow x=y=z\\ \Rightarrow A=\left(1+1\right).\left(1+1\right).\left(1+1\right)=8\)
X
21 tháng 1 2019
Đặt : \(\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{z+x}=M\)
\(\Rightarrow\left(x+y+z\right).M=\frac{1}{672}.2017\)
\(\Rightarrow1+\frac{z}{x+y}+1+\frac{x}{y+z}+1+\frac{y}{z+x}=\frac{2016}{672}+\frac{1}{672}\)
\(\Rightarrow3+\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=3+\frac{1}{672}\)
\(\Rightarrow\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=\frac{1}{672}\)
21 tháng 1 2019
Nhân cả 2 vế với \(x+y+z\),ta được:
\(\left(x+y+z\right)\left(\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{z+x}\right)=\frac{1}{672}\cdot2017\)
\(\Rightarrow\frac{x+y+z}{x+y}+\frac{x+y+z}{y+z}+\frac{x+y+z}{z+x}=\frac{2017}{672}\)
\(\Rightarrow3+\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}=\frac{2017}{672}\)
\(\Rightarrow C=\frac{1}{672}\)
KT
27 tháng 4 2018
Áp dụng tính chất dãy tỉ số bằng nhau ta có:
\(\frac{x-y-z}{x}=\frac{-x+y-z}{y}=\frac{-x-y+z}{z}=\frac{x-y-z-x+y-z-x-y+z}{x+y+z}\)\(=\frac{-\left(x+y+z\right)}{x+y+z}\)
Nếu \(x+y+z=0\)thì \(\hept{\begin{cases}x+y=-z\\y+z=-x\\z+x=-y\end{cases}}\)
\(A=\left(1+\frac{y}{x}\right)\left(1+\frac{z}{y}\right)\left(1+\frac{x}{z}\right)\)
\(=\frac{x+y}{x}.\frac{y+z}{y}.\frac{z+x}{z}\)
\(=\frac{-z}{x}.\frac{-x}{y}.\frac{-y}{z}=-1\)
Nếu \(x+y+z\ne0\)thì \(\frac{x-y-z}{x}=\frac{-x+y-z}{y}=\frac{-x-y+z}{z}=-1\)
suy ra: \(\frac{x-y-z}{x}=-1\) \(\Rightarrow\) \(x-y-z=-x\) \(\Rightarrow\) \(y+z=2x\)
\(\frac{-x+y-z}{y}=-1\) \(-x+y-z=-y\) \(x+z=2y\)
\(\frac{-x-y+z}{z}=-1\) \(-x-y+z=-z\) \(x+y=2z\)
\(A=\left(1+\frac{y}{x}\right)\left(1+\frac{z}{y}\right)\left(1+\frac{x}{z}\right)\)
\(=\frac{x+y}{x}.\frac{y+z}{y}.\frac{x+z}{z}\)
\(=\frac{2z}{x}.\frac{2x}{y}.\frac{2y}{z}=8\)
30 tháng 1 2017
+ Nếu x + y + z = 0 => x + y = -z; y + z = -x; x + z = -y
A = (1 + y/x)(1 + z/y)(1 + x/z)
A = (x+y)/x . (y+z)/y . (x+z)/z
A = -z/x . (-x)/y . (-y)/z = -1
+ Nếu x + y + z khác 0
x-y-z/x = -x+y-z/y = -x-y+z/z
<=> 1 - (y+z)/x = 1 - (x+z)/y = 1 - (x+y)/z
<=> y+z/x = x+z/y = x+y/z
Áp dụng t/c của dãy tỉ số = nhau ta có:
y+z/x = x+z/y = x+y/z = 2(x+y+z)/x+y+z = 2
A = (x+y)/x . (y+z)/y . (x+z)/z = 8
13 tháng 2 2019
\(\hept{\begin{cases}x+y+z=2010\\\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{2010}\end{cases}\Rightarrow\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{x+y+z}}\)
\(\Rightarrow\left(\frac{1}{x}+\frac{1}{y}\right)+\left(\frac{1}{z}-\frac{1}{x+y+z}\right)=0\)
\(\Leftrightarrow\frac{x+y}{xy}+\frac{x+y+z-z}{z\left(x+y+z\right)}=0\)
\(\Leftrightarrow\left(x+y\right)\left[\frac{1}{xy}+\frac{1}{z\left(x+y+z\right)}\right]=0\)
\(\Leftrightarrow\left(x+y\right)\left[\frac{z\left(x+y+z\right)+xy}{xyz\left(x+y+z\right)}\right]=0\)
\(\Leftrightarrow\left(x+y\right)\left[\frac{zx+zy+z^2+xy}{xyz\left(x+y+z\right)}\right]=0\)
\(\Leftrightarrow\left(x+y\right)\left[\frac{z\left(x+z\right)+y\left(z+x\right)}{xyz\left(x+y+z\right)}\right]=0\)
\(\Leftrightarrow\left(x+y\right)\left[\frac{\left(x+z\right)\left(z+y\right)}{xyz\left(x+y+z\right)}\right]=0\)
\(\Leftrightarrow\frac{\left(x+y\right)\left(x+z\right)\left(z+y\right)}{xyz\left(x+y+z\right)}=0\)
\(\Leftrightarrow\left(x+y\right)\left(x+z\right)\left(z+y\right)=0\)
<=> x+y = 0 hoặc x+z=0 hoặc z+y=0
<=> x = -y hoặc x = -z hoặc z = -y
\(\Rightarrow P=\left(x^{2007}+y^{2007}\right)\left(y^{2009}+z^{2009}\right)\left(z^{2009}+x^{2009}\right)=0\)
ta có: \(\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{x+z}=\frac{1}{90}.\)
\(\Rightarrow2007.\left(\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{x+z}\right)=2007\cdot\frac{1}{90}\)
\(\frac{2007}{x+y}+\frac{2007}{y+z}+\frac{2007}{x+z}=\frac{223}{10}\)
mà x+y+z = 2007
\(\Rightarrow\frac{x+y+z}{x+y}+\frac{x+y+z}{y+z}+\frac{x+y+z}{x+z}=\frac{223}{10}\)
\(1+\frac{z}{x+y}+1+\frac{x}{y+z}+1+\frac{y}{x+z}=\frac{223}{10}\)
\(\Rightarrow\frac{z}{x+y}+\frac{x}{y+z}+\frac{y}{x+z}=\frac{223}{10}-3=\frac{193}{10}\)