Áp dụng t/c dãy tỉ số bằng nhau

\(\dfrac{a}{2013}=\dfrac{b}{2012}=\dfrac{c}{2011}=\dfrac{a-c}{2}=\dfrac{a-b}{1}=\dfrac{b-c}{1}\\ \Rightarrow a-c=2\left(a-b\right)=2\left(b-c\right)\)

\(\Rightarrow H=\dfrac{\left[2\left(a-b\right)\right]^4}{\left(a-b\right)^2\left(a-b\right)^2}=\dfrac{16\left(a-b\right)^4}{\left(a-b\right)^4}=16\)

Áp dụng tính chất dãy tỉ số bằng nhau ta có:

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

\(\Rightarrow\left\{{}\begin{matrix}b+c=2a\\c+a=2b\\a+b=2c\end{matrix}\right.\)

\(\Rightarrow P=\dfrac{b+c}{a}+\dfrac{c+a}{b}+\dfrac{a+b}{c}=\dfrac{2a}{a}+\dfrac{2b}{b}+\dfrac{2c}{c}=2+2+2=6\)

P=
\(\dfrac{b+c}{a}+\dfrac{c+a}{b}+\dfrac{a+b}{c}=\dfrac{a}{b+c}.\left(\dfrac{b+c}{a}+\dfrac{c+a}{b}+\dfrac{a+b}{c}\right):\left(\dfrac{a}{b+c}\right)=\left(\dfrac{b+c}{a}.\dfrac{a}{b+c}+\dfrac{c+a}{b}.\dfrac{a}{b+c}+\dfrac{a+b}{c}.\dfrac{a}{b+c}\right):\dfrac{a}{b+c}=\left(\dfrac{b+c}{a}.\dfrac{a}{b+c}+\dfrac{c+a}{b}.\dfrac{b}{c+a}+\dfrac{a+b}{c}.\dfrac{c}{a+b}\right):\dfrac{a}{b+c}=\left(1+1+1\right):\dfrac{a}{b+c}=3.\dfrac{b+c}{a}=\dfrac{3b+3c}{a}\)

Lời giải:

Bổ sung điều kiện $a,b,c$ không thể đồng thời bằng $0$

Từ đkđb suy ra:
\(\frac{6(10a-15b)}{2007.6}=\frac{15(6b-10c)}{15.2008}=\frac{10(15c-6a)}{10.2009}\)

Áp dụng tính chất dãy tỉ số bằng nhau:

\(\frac{6(10a-15b)}{2007.6}=\frac{15(6b-10c)}{15.2008}=\frac{10(15c-6a)}{10.2009}=\frac{6(10a-15b)+15(6b-10c)+10(15c-6a)}{2007.6+15.2008+10.2009}=0\)

\(\Rightarrow 10a-15b=6b-10c=15c-6a=0\)

\(\Leftrightarrow 10a=15b; 6b=10c; 15c=6a\Leftrightarrow \frac{a}{15}=\frac{b}{10}=\frac{c}{6}\)

Đặt $\frac{a}{15}=\frac{b}{10}=\frac{c}{6}=k$ thì: $a=15k, b=10k, c=6k$

Vì $a,b,c$ không thể đồng thời bằng $0$ nên $k\neq 0$

Khi đó: 

$P=\frac{15k.10k+10k.6k+15k.6k}{(15k)^2+(10k)^2+(6k)^2}$

$=\frac{300k^2}{361k^2}=\frac{300}{361}$

Áp dụng tính chất dãy tỉ số bằng nhau:

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

\(\Rightarrow\left\{{}\begin{matrix}2a=b+c\\2b=a+c\\2c=a+b\end{matrix}\right.\)

\(\Rightarrow P=\dfrac{\left(a+b\right)^3}{c^3}+\dfrac{\left(b+c\right)^3}{a^3}+\dfrac{\left(c+a\right)^3}{b^3}=\left(\dfrac{a+b}{c}\right)^3+\left(\dfrac{b+c}{a}\right)^3+\left(\dfrac{c+a}{b}\right)^3=\left(\dfrac{2c}{c}\right)^3+\left(\dfrac{2a}{a}\right)^3+\left(\dfrac{2b}{b}\right)^3=2^3+2^3+2^3=24\)

\(a,A=\dfrac{9x}{x}:\left[\dfrac{x\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)^2}-\dfrac{4\sqrt{x}}{\sqrt{x}\left(\sqrt{x}-2\right)}\right]\\ A=9:\left(\dfrac{x}{\sqrt{x}-2}-\dfrac{4}{\sqrt{x}-2}\right)=9:\dfrac{x-4}{\sqrt{x}-2}\\ A=\dfrac{9\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}=\dfrac{9}{\sqrt{x}+2}\\ b,x=11+2\sqrt{30}\Leftrightarrow\sqrt{x}=\sqrt{6}+\sqrt{5}\\ \Leftrightarrow A=\dfrac{9}{\sqrt{6}+\sqrt{5}+2}=\dfrac{9\left(\sqrt{6}+\sqrt{5}-2\right)}{7+2\sqrt{30}}\\ \Leftrightarrow A=\dfrac{9\left(\sqrt{6}+\sqrt{5}-2\right)\left(2\sqrt{30}-7\right)}{71}\)

\(c,A+\sqrt{x}=\dfrac{9}{\sqrt{x}+2}+\sqrt{x}=\dfrac{9}{\sqrt{x}+2}+\left(\sqrt{x}+2\right)-2\\ A+\sqrt{x}\ge2\sqrt{\dfrac{9\left(\sqrt{x}+2\right)}{\sqrt{x}+2}}-2=2\sqrt{9}-2=4\left(đpcm\right)\)