a) \(\dfrac{x}{33}=4\Rightarrow x=33.4=132\)

b) \(\dfrac{5}{x}=\dfrac{1}{2}\Rightarrow x=5.2=10\)

a) \(\dfrac{22}{55}+\dfrac{25}{125}+\dfrac{1020}{5100}=\dfrac{2}{5}+\dfrac{1}{5}+\dfrac{1}{5}=\dfrac{4}{5}\)

b) \(\dfrac{22}{77}+\dfrac{56}{98}+\dfrac{25}{105}=\dfrac{2}{7}+\dfrac{4}{7}+\dfrac{5}{21}=\dfrac{6+12+5}{21}=\dfrac{23}{21}\)

Áp dụng bđt: \(\dfrac{1}{x+y}\le\dfrac{1}{4}\left(\dfrac{1}{x}+\dfrac{1}{y}\right)\left(1\right)\)

\(\dfrac{1}{2a+b+c}=\dfrac{1}{\left(a+b\right)+\left(a+c\right)}\le\dfrac{1}{4}\left(\dfrac{1}{a+b}+\dfrac{1}{a+c}\right)\)

\(\Rightarrow P\le\dfrac{1}{16}\left[\left(\dfrac{1}{a+b}+\dfrac{1}{a+c}\right)^2+\left(\dfrac{1}{a+b}+\dfrac{1}{b+c}\right)^2+\left(\dfrac{1}{b+c}+\dfrac{1}{a+c}\right)^2\right]\)\(\Rightarrow16P\le\dfrac{2}{\left(a+b\right)^2}+\dfrac{2}{\left(b+c\right)^2}+\dfrac{2}{\left(a+c\right)^2}+\dfrac{2}{\left(a+b\right)\left(b+c\right)}+\dfrac{2}{\left(a+b\right)\left(b+c\right)}+\dfrac{2}{\left(b+c\right)\left(c+a\right)}\)

Áp dụng: \(x^2+y^2+z^2\ge xy+yz+xz\left(2\right)\) với a+b=x,b+c=y,c+a=z

\(\Rightarrow16P\le\dfrac{4}{\left(a+b\right)^2}+\dfrac{4}{\left(b+c\right)^2}+\dfrac{4}{\left(c+a\right)^2}\)

Ta có: \(\dfrac{1}{\left(a+b\right)^2}\le4.16.\left(\dfrac{1}{a}+\dfrac{1}{b}\right)^2\)(do (1))

\(\Rightarrow16P\le\dfrac{1}{4}.16\left[\left(\dfrac{1}{a}+\dfrac{1}{b}\right)^2+\left(\dfrac{1}{b}+\dfrac{1}{c}\right)^2+\left(\dfrac{1}{c}+\dfrac{1}{a}\right)^2\right]=\dfrac{1}{4}\left(\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}+\dfrac{2}{ab}+\dfrac{2}{bc}+\dfrac{2}{ca}\right)\le\dfrac{1}{4}.4.\left(\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}\right)=3\)(do(2) và \(\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}=3\))

\(\Rightarrow P\le\dfrac{3}{16}\)

\(ĐTXR\Leftrightarrow a=b=c=1\)

\(x^2+10y^2-2xy+6x+1=0\Leftrightarrow\left(x-y\right)^2+\left(3y+1\right)^2=0\)

Vì \(\left(x-y\right)^2\ge0,\left(3y+1\right)^2\ge0\)

\(\Rightarrow\left\{{}\begin{matrix}x-y=0\\3y+1=0\end{matrix}\right.\)\(\Rightarrow x=y=\dfrac{-1}{3}\)

a) Các số tự nhiên x cần tìm là: 5;6;7;8

b) 290<x<705

Các số tự nhiên x cần tìm là: 300;400;500;600;700

\(\left(-1\right).\left(-1\right)^2.\left(-1\right)^3...\left(-1\right)^{2010}.\left(-1\right)^{2011}=\left[\left(-1\right).\left(-1\right)^3.\left(-1\right)^5...\left(-1\right)^{2011}\right].\left[\left(-1\right)^2.\left(-1\right)^4...\left(-1\right)^{2010}\right]=1.1=1\)