Ta có: \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{x+y+z}\)

\(\Leftrightarrow\frac{xy+yz+zx}{xyz}=\frac{1}{x+y+z}\)

\(\Leftrightarrow\left(xy+yz+zx\right)\left(x+y+z\right)=xyz\)

\(\Leftrightarrow x^2y+xy^2+y^2z+yz^2+z^2x+zx^2+3xyz-xyz=0\)

\(\Leftrightarrow\left(x^2y+xy^2\right)+\left(yz^2+z^2x\right)+\left(zx^2+2xyz+y^2z\right)=0\)

\(\Leftrightarrow xy\left(x+y\right)+z^2\left(x+y\right)+z\left(x+y\right)^2=0\)

\(\Leftrightarrow\left(x+y\right)\left(xy+z^2+yz+zx\right)=0\)

\(\Leftrightarrow\left(x+y\right)\left(y+z\right)\left(z+x\right)=0\)

=> x = -y hoặc y = -z hoặc z = -x

Không mất tổng quát giả sử x = -y, khi đó:

\(\frac{1}{x^{2015}}+\frac{1}{y^{2015}}+\frac{1}{z^{2015}}=-\frac{1}{y^{2015}}+\frac{1}{y^{2015}}+\frac{1}{z^{2015}}=\frac{1}{z^{2015}}\)

\(\frac{1}{x^{2015}+y^{2015}+z^{2015}}=\frac{1}{-y^{2015}+y^{2015}+z^{2015}}=\frac{1}{z^{2015}}\)

\(\Rightarrow\frac{1}{x^{2015}}+\frac{1}{y^{2015}}+\frac{1}{z^{2015}}=\frac{1}{x^{2015}+y^{2015}+z^{2015}}\)

Có \(\frac{1}{x}+\frac{1}{y}=\frac{1}{2015}\)

<=> \(\frac{x+y}{xy}=\frac{1}{2015}=>xy=2015\left(x+y\right)\)

Có P²=\(\frac{x+y}{x-2015+y-2015+2\sqrt{xy-2015\left(x+y\right)+2015^2}}\) =\(\frac{x+y}{\left(x+y\right)-4030+2\sqrt{xy-xy+2015^2}}\)( vì 2015(x+y)=xy)

= \(\frac{x+y}{x+y-4030+2\sqrt{2015^2}}=\frac{x+y}{x+y-4030+2.2015}=\frac{x+y}{x+y}\)=1

=> P=1(vì P>0)

\(\frac{x-1}{2015}-\frac{1}{2015}=\frac{10-2x}{2015}\)

\(\Rightarrow x-1-1=10-2x\)

\(\Rightarrow x-2=10-2x\)

\(\Rightarrow-2x+x=10+2\)

\(\Rightarrow-x=12\Rightarrow x=-12\)

\(\Rightarrow\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2015}\right)x=\left(\frac{2016}{1}-1\right)+\left(\frac{2017}{2}-1\right)+...+\left(\frac{4030}{2015}-1\right)\)

\(\Rightarrow\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2015}\right)x=\frac{2015}{1}+\frac{2015}{2}+...+\frac{2015}{2015}\)

\(\Rightarrow\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2015}\right)x=2015.\left(1+\frac{1}{2}+...+\frac{1}{2015}\right)\)

=> x = 2015

\(\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2015}\right).x+2015=\frac{2016}{1}+\frac{2017}{2}+\frac{2018}{3}+...+\frac{4030}{2015}\)

\(\Rightarrow\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2015}\right).x=\left(\frac{2016}{1}-1\right)+\left(\frac{2017}{2}-1\right)+...+\left(\frac{4030}{2015}-1\right)\)

\(\Rightarrow\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2015}\right).x=\frac{2015}{1}+\frac{2015}{2}+...+\frac{2015}{2015}=2015.\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2015}\right)\)\(\Rightarrow x=2015\)

giải rồi mà cũng hỏi                

\(\frac{x-4}{2015}-\frac{1}{2015}=\frac{10-2x}{2015}\)

\(\Rightarrow\frac{x-4}{2015}-\frac{10-2x}{2015}=\frac{1}{2015}\)

\(\Rightarrow\frac{x-4-\left(10-2x\right)}{2015}=\frac{1}{2015}\)

\(\Rightarrow\frac{\left(x+2x\right)-\left(4+10\right)}{2015}=\frac{1}{2015}\)

\(\Rightarrow\frac{3x-14}{2015}=\frac{1}{2015}\)

\(\Rightarrow\left(3x-14\right).2015=2015\)

\(\Rightarrow3x-14=1\) ( bớt cả 2 vế đi 2015 lần )

\(\Rightarrow3x=15\)

\(\Rightarrow x=5\)

Vậy \(x=5\)

X=5

\(\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2015}\right)x+2015=\frac{2016}{1}+\frac{2017}{2}+...+\frac{4030}{2015}\)

\(\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2015}\right)x=\left(\frac{2016}{1}-1\right)+\left(\frac{2017}{2}-1\right)+...+\left(\frac{4030}{2015}-1\right)\)

\(\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2015}\right)x=\frac{2015}{1}+\frac{2015}{2}+...+\frac{2015}{2015}\)

\(\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2015}\right)x=2015.\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2015}\right)\)

\(\Rightarrow x=2015\)

Bạn có thể tham khảo nhé!^-^

Nhi giải nhanh đi