vì giá trị tuyệt đối không nhận giá trị âm nên

x-2015=0;x-2016=0;y2017=0;y-2018=0

=>x=2015;x=2016;y=2017;y=2018

Vì x và y không nhận hai giá trị cùng một lúc nên x y không tồn tại

Vì \(x^{2015}+y^{2015}=x^{2016}+y^{2016}=x^{2017}+y^{2017}\)

\(\Rightarrow x=y=1\) hoặc \(x=y=0\)

Với \(x=y=1\)

\(S=2018\left(1^{2018}+1^{2018}\right)\)

\(S=2018.2\)

\(S=4036\)

Với \(x=y=0\)

\(S=2018\left(0^{2018}+0^{2018}\right)\)

\(S=0\)

Áp dụng bđt: \(\left|a\right|+\left|b\right|\ge\left|a+b\right|\) ta có:

\(VT=\left|x-2015\right|+\left|x-2016\right|+\left|y-2017\right|+\left|x-2018\right|\)

\(VT=\left|x-2015\right|+\left|2018-x\right|+\left|x-2016\right|+\left|y-2017\right|\)

\(VT\ge\left|x-2015+2018-x\right|+\left|x-2016\right|+\left|y-2017\right|\)

\(VT\ge3+\left|x-2016\right|+\left|y-2017\right|\ge3\)

\(VT\ge VP\)

Dấu "=" khi: \(\left\{{}\begin{matrix}2015\le x\le2018\\x=2016\\y=2017\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=2016\\y=2017\end{matrix}\right.\)

Ta có: \(\frac{x-2019}{2018}+\frac{x-2018}{2017}=\frac{x-2017}{2016}+\frac{x-2016}{2015}\)

\(\Leftrightarrow\left(\frac{x-2019}{2018}+1\right)+\left(\frac{x-2018}{2017}+1\right)=\left(\frac{x-2017}{2016}+1\right)+\left(\frac{x-2016}{2015}+1\right)\)

\(\Leftrightarrow\frac{x-1}{2018}+\frac{x-1}{2017}=\frac{x-1}{2016}+\frac{x-1}{2015}\)

\(\Leftrightarrow\frac{x-1}{2018}+\frac{x-1}{2017}-\frac{x-1}{2016}-\frac{x-1}{2015}=0\)

\(\Leftrightarrow\left(x-1\right)\left(\frac{1}{2018}+\frac{1}{2017}-\frac{1}{2016}-\frac{1}{2015}\right)=0\)

\(\Leftrightarrow x-1=0\)( vì \(\frac{1}{2018}+\frac{1}{2017}-\frac{1}{2016}-\frac{1}{2015}\ne0\))

\(\Leftrightarrow x=1\)

Vạy x=1

trừ mỗi vế cho 2 rồi tách -2 thành -1và -1

X=1 nhé

     \(\frac{x+2014}{2015}+\frac{x+2015}{2016}=\frac{x+2016}{2017}+\frac{x+2017}{2018}\)

\(\Leftrightarrow\)\(\frac{x+2014}{2015}-1+\frac{x+2015}{2016}-1=\frac{x+2016}{2017}-1+\frac{x+2017}{2018}-1\)

\(\Leftrightarrow\)\(\frac{x-1}{2015}+\frac{x-1}{2016}=\frac{x-1}{2017}+\frac{x-1}{2018}\)

\(\Leftrightarrow\)\(\left(x-1\right)\left(\frac{1}{2015}+\frac{1}{2016}-\frac{1}{2017}-\frac{1}{2018}\right)=0\)

\(\Leftrightarrow\)\(x-1=0\)   ( do  1/2015 + 1/2016 - 1/2017 - 1/2018 # 0 )

\(\Leftrightarrow\) \(x=1\)