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Đặt : \(\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{z+x}=M\)
\(\Rightarrow\left(x+y+z\right).M=\frac{1}{672}.2017\)
\(\Rightarrow1+\frac{z}{x+y}+1+\frac{x}{y+z}+1+\frac{y}{z+x}=\frac{2016}{672}+\frac{1}{672}\)
\(\Rightarrow3+\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=3+\frac{1}{672}\)
\(\Rightarrow\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=\frac{1}{672}\)
Nhân cả 2 vế với \(x+y+z\),ta được:
\(\left(x+y+z\right)\left(\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{z+x}\right)=\frac{1}{672}\cdot2017\)
\(\Rightarrow\frac{x+y+z}{x+y}+\frac{x+y+z}{y+z}+\frac{x+y+z}{z+x}=\frac{2017}{672}\)
\(\Rightarrow3+\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}=\frac{2017}{672}\)
\(\Rightarrow C=\frac{1}{672}\)
làm lần lượt nhá,dài dòng quá khó coi.ahihihi!
\(\frac{1-\frac{1}{\sqrt{49}}+\frac{1}{49}-\frac{1}{7\left(\sqrt{7}\right)^2}}{\frac{\sqrt{64}}{2}-\frac{4}{7}+\left(\frac{2}{7}\right)^2-\frac{4}{343}}=\frac{1-\frac{1}{7}+\frac{1}{49}-\frac{1}{343}}{4-\frac{4}{7}+\frac{4}{49}-\frac{4}{343}}\)
\(=\frac{1-\frac{1}{7}+\frac{1}{49}-\frac{1}{343}}{4\left(1-\frac{1}{7}+\frac{1}{49}-\frac{1}{343}\right)}=\frac{1}{4}\)
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
\(\Rightarrow\frac{x+y+z}{x+y}+\frac{x+y+z}{y+z}+\frac{x+y+z}{x+z}=\frac{223}{10}\)
\(1+\frac{z}{x+y}+1+\frac{x}{y+z}+1+\frac{y}{x+z}=\frac{223}{10}\)
\(\Rightarrow\frac{z}{x+y}+\frac{x}{y+z}+\frac{y}{x+z}=\frac{223}{10}-3=\frac{193}{10}\)
Áp dụng t/c của dãy tỉ số bằng nhau, ta có:
\(\frac{y+z+1}{x}=\frac{x+z+2}{y}=\frac{x+y-3}{z}=\frac{1}{x+y+z}=\frac{y+z+1+x+z+2+x+y-3}{x+y+z}=\frac{2\left(x+y+z\right)}{x+y+z}=2\)(vì x + y + z khác 0)
=> \(\frac{1}{x+y+z}=2\) => x + y + z = 1/2
=> \(\hept{\begin{cases}\frac{y+z+1}{x}=2\\\frac{x+z+2}{y}=2\\\frac{x+y-3}{z}=2\end{cases}}\) => \(\hept{\begin{cases}y+z+1=2x\\x+z+2=2y\\x+y-3=2z\end{cases}}\) => \(\hept{\begin{cases}3x=x+y+z+1\\3y=x+y+z+2\\3z=x+y+z-3\end{cases}}\)=> \(\hept{\begin{cases}3x=\frac{3}{2}\\3y=\frac{5}{2}\\3z=-\frac{5}{2}\end{cases}}\)=> \(\hept{\begin{cases}x=\frac{1}{2}\\y=\frac{5}{6}\\z=-\frac{5}{6}\end{cases}}\)
Khi đó: A = \(2016\cdot\frac{1}{2}+\left(\frac{5}{6}\right)^{2017}-\left(\frac{5}{6}\right)^{2017}=1008\)
Ta có \(\frac{y+z+1}{x}=\frac{x+z+2}{y}=\frac{x+y-3}{z}=\frac{1}{x+y+z}=\frac{y+z+1+x+z+2+x+y-3}{x+y+z}\)
\(=\frac{2\left(x+y+z\right)}{x+y+z}=2\)
Khi đó \(\frac{1}{x+y+z}=2\Rightarrow x+y+z=\frac{1}{2}\)
Lại có \(\frac{y+z+1}{x}=2\Rightarrow y+z+1=2x\Rightarrow x+y+z+1=3x\Rightarrow\frac{1}{2}+1=3x\Rightarrow3x=\frac{3}{2}\)
=> x = 1/2
Lại có \(\frac{x+z+2}{y}=2\Rightarrow x+z+2=2y\Rightarrow x+y+z+2=3y\Rightarrow\frac{1}{2}+2=3y\Rightarrow3y=\frac{5}{2}\)
=> y = 5/6
Lại có x + y + z = 1/2
=> 1/2 + 5/6 + z = 1/2
=> 5/6 + z = 0
=> z = -5/6
Khi đó A = 2016X + y2017 + z2017
= 2016.1/2 + (5/6)2017 - (5/6)2017
= 1008
Vậy A = 1008
Theo đề bài để tồn tại phân số: \(\frac{1}{x+y+z}\) ta có: \(x+y+z\ne0\)
Áp dụng tính chất dãy tỉ số bằng nhau ta có:
\(\frac{y+z+1}{x}=\frac{x+z+2}{y}=\frac{x+y-3}{z}=\frac{2\left(x+y+z\right)}{x+y+z}=2\)
\(\Rightarrow\frac{1}{x+y+z}=2\Leftrightarrow x+y+z=\frac{1}{2}\Leftrightarrow\hept{\begin{cases}x+y=\frac{1}{2}-z\\y+z=\frac{1}{2}-x\\z+x=\frac{1}{2}-y\end{cases}}\)
Thay vào đề bài ta có: \(\frac{\frac{1}{2}-x+1}{x}=\frac{\frac{1}{2}-y+2}{y}=\frac{\frac{1}{2}-z-3}{z}=2\)
Dễ dàng tìm được x;y;z rồi thay vào b thức
\(\dfrac{y+z-x}{x}=\dfrac{z+x-y}{y}=\dfrac{x+y-z}{z}\\ \Rightarrow\dfrac{y+z-x}{x}+2=\dfrac{z+x-y}{y}+2=\dfrac{x+y-z}{z}+2\\ \Rightarrow\dfrac{x+y+z}{x}=\dfrac{x+y+z}{y}=\dfrac{x+y+z}{z}\\ \Rightarrow x=y=z\\ \Rightarrow A=\left(1+1\right).\left(1+1\right).\left(1+1\right)=8\)
\(\frac{y+z+1}{x}=\frac{x+z+2}{y}=\frac{x+y-3}{z}=\frac{1}{x+y+z}\)
Áp dụng tính chất dãy tỉ số bằng nhau ta có:
\(\frac{y+z+1}{x}=\frac{x+z+2}{y}=\frac{x+y-3}{z}=\frac{1}{x+y+z}=\frac{y+z+1+x+z+2+x+y-3}{x+y+z}=2\)
\(\Rightarrow\hept{\begin{cases}y+z+1=2x\\x+z+2=2y\\x+y+z=\frac{1}{2}\end{cases}}\Leftrightarrow\hept{\begin{cases}x=\frac{1}{2}\\y=\frac{5}{6}\\z=-\frac{5}{6}\end{cases}}\)
\(A=2016x+y^{2017}+z^{2017}=2016.\frac{1}{2}+\left(\frac{5}{6}\right)^{2017}+\left(-\frac{5}{6}\right)^{2017}=1008\)
Ta có : \(\left(x+y+z\right)\left(\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{z+x}\right)=\) = \(\frac{2017}{672}\)
\(\Leftrightarrow\frac{x+y+z}{x+y}+\frac{x+y+z}{y+z}+\frac{x+y+z}{z+x}=\)\(\frac{2017}{672}\)
\(\Leftrightarrow1+\frac{z}{x+y}+1+\frac{x}{y+z}+1+\frac{z}{z+x}\)= \(\frac{2017}{672}\)
\(\Rightarrow A=\frac{2017}{672}-3\)