Bài 1:

Đặt x-2009=y. Khi đó phương trình đã cho trở thành:

\(\frac{y^2-y\left(y-1\right)+\left(y-1\right)^2}{y^2+y\left(y-1\right)+\left(y-1\right)^2}=\frac{19}{49}\)

\(\Leftrightarrow4y^2-4y-15=0\)

\(\Leftrightarrow\)(2y-5).(2y+3)=0

\(\Leftrightarrow\left[\begin{matrix}y=2,5\\y=-1,5\end{matrix}\right.\)

Thay y=x-2009. Ta được: \(\left[\begin{matrix}x=2009+2,5=2011,5\\x=2009-1,5=2007,5\end{matrix}\right.\)

Vậy x=2011,5 hoặc x=2007,5

A=/x-2008/+/2009-x/+/y-2010/+/x-2011/+2011

≥/x-2008+2009-x/+/y-2010/+/x-2011/+2011

= /y-2010/+/x-2011/+2012≥2012

Dau bang xay ra khi : \(\left\{{}\begin{matrix}y-2010=0\\x-2011=0\end{matrix}\right.\)

<=> \(\left\{{}\begin{matrix}y=2010\\x=2011\end{matrix}\right.\)

Vay GTNN cua A=2012 khi \(\left\{{}\begin{matrix}x=2011\\y=2010\end{matrix}\right.\)

Để ý tử và mẫu là hằng đẳng thức

đặt 2009-x=a,x-2010=b

suy ra a^2+ab+b^2/a^2-ab+b^2=19/49 

suy ra 49(a^2+ab+b^2)=19(a^2-ab+b^2)

49a^2+49ab+49b^2=19a^2-19ab+19b^2

30a^2+68ab+30b^2=0

30a^2+50ab+18ab+30b^2=0

10a(3a+5b)+6b(3a+5b)=0

(3a+5b)(10a+6b)=0

suy ra 3a+5b=0 hoặc 10a+6b=0 

thế vào lại rồi tìm x 

đề sai rồi th ngu

Đặt \(\left\{{}\begin{matrix}x-2010=a\\2009-x=b\end{matrix}\right.\)

Theo đề bài ta có:

\(\dfrac{\left(2009-x\right)^2+\left(2009-x\right)\left(x-2010\right)+\left(x-2010\right)^2}{\left(2009-x\right)^2-\left(2009-x\right)\left(x-2010\right)+\left(x-2010\right)^2}=\dfrac{19}{49}\)

\(\Leftrightarrow\dfrac{b^2+ab+a^2}{b^2-ab+a^2}=\dfrac{19}{49}\)

\(\Leftrightarrow19\left(b^2-ab+a^2\right)=49\left(b^2+ab+a^2\right)\)
\(\Leftrightarrow19b^2-19ab+19a^2-49b^2-49ab-49a^2=0\)

\(\Leftrightarrow-30a^2-68ab-30b^2=0\)

\(\Leftrightarrow-2\left(15a^2+34ab+15b^2\right)=0\)

\(\Leftrightarrow15a^2+34ab+15b^2=0\)

\(\Leftrightarrow15a^2+25ab+9ab+15b^2=0\)

\(\Leftrightarrow5a\left(3a+5b\right)+3b\left(3a+5b\right)=0\)

\(\Leftrightarrow\left(3a+5b\right)\left(5a+3b\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}3a+5b=0\\5a+3b=0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}3\left(x-2010\right)+5\left(2009-x\right)=0\\5\left(x-2010\right)+3\left(2009-x\right)=0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}3x-6030+10045-5x=0\\5x-10050+6027-3x=0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}-2x+4015=0\\2x-4023=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}-2x=-4015\\2x=4023\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{-4015}{-2}=2007,5\\x=\dfrac{4023}{2}=2011,5\end{matrix}\right.\)

Vậy \(\left[{}\begin{matrix}x=2007,5\\x=2011,5\end{matrix}\right.\)

Đặt a=(2009-x)²

b=(x-2010)²

Theo đề bài ta có

\(\dfrac{\text{a^2+ab+b^2}}{a^2-ab+b^2}=\dfrac{19}{49}\)

\(\text{49(a^2+ab+b^2)}=19\left(a^2-ab+b^2\right)\)

\(\text{30a^2+68ab+30b^2=0}\)

\(\text{15a^2+34ab+15b^2=0}\)

\(\text{15a^2+9ab+25ab+15b^2=0}\)

\(\text{3a(5a+3b)+5(3b+5a)=0}\)

\(\text{(5a+3b)(3a+5b)=0}\)

\(\left[{}\begin{matrix}3a+5b=0\\3b+5a=0\end{matrix}\right.\)

\(\left[{}\begin{matrix}3\left(2009-x\right)=5\left(x-2010\right)\\5\left(2009-x\right)=3\left(x-2010\right)\end{matrix}\right.\)

\(-8x=-6030-10045\) hay \(8x=-10050-6027\)

\(x\simeq2009\),375 hay \(x\simeq2009,625\)

Đặt

Theo đề bài ta có:

Vậy

Đặt x-2009=a\(\Leftrightarrow\dfrac{\left(x-2009\right)^2-\left(x-2009\right)\left(x-2010\right)+\left(x-2010\right)^2}{\left(x-2009\right)^2+\left(x-2009\right)\left(x-2010\right)+\left(x-2010\right)^2}=\dfrac{19}{49}\)

\(\Leftrightarrow\dfrac{a^2-a\left(a-1\right)+\left(a-1\right)^2}{a^2+a\left(a-1\right)+\left(a-1\right)^2}=\dfrac{19}{49}\)

\(\Leftrightarrow\dfrac{a^2-a^2+a+a^2-2a+1}{a^2+a^2-a+a^2-2a+1}=\dfrac{19}{49}\)

=>\(\dfrac{a^2-a+1}{3a^2-3a+1}=\dfrac{19}{49}\)

=>49a^2-49a+49-57a^2+57a-19=0

=>-8a^2+8a+30=0

=>a=5/2 hoặc a=-3/2

=>x-2009=5/2 hoặc x-2009=-3/2

=>x=4023/2 hoặc x=4015/2

A=|x-2008|+|2009-x|+|y-2010|+|x-2011|+2011

≥|x-2008+2009-x|+|y-2010|+|x-2011|+2011

= |y-2010|+|x-2011|+2012≥2012

Dấu = xảy ra khi : {y−2010=0x−2011=0{y−2010=0x−2011=0

<=> {y=2010x=2011{y=2010x=2011

Vay GTNN cua A=2012 khi {x=2011;y=2010