\(P\ge\left(a+b+c\right)^2\left(\dfrac{1}{a^2+b^2+c^2}+\dfrac{9}{ab+bc+ca}\right)\)

\(P\ge\left(a+b+c\right)^2\left(\dfrac{1}{a^2+b^2+c^2}+\dfrac{1}{ab+bc+ca}+\dfrac{1}{ab+bc+ca}+\dfrac{7}{ab+bc+ca}\right)\)

\(P\ge\left(a+b+c\right)^2\left(\dfrac{9}{a^2+b^2+c^2+2ab+2bc+2ca}+\dfrac{7}{\dfrac{1}{3}\left(a+b+c\right)^2}\right)=30\)

\(P_{min}=30\) khi \(a=b=c\)

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áp dụng bất đẳng thức: 1+b²>=2b. tương tự.....

ad bđt cauchy: a/b+b/c+c/a>=3∛a/b.b/c.c/a=3

P>=\(\dfrac{2ab}{bc}\)+\(\dfrac{2bc}{ca}\)+\(\dfrac{2ca}{ab}\) =2(\(\dfrac{a}{b}\)+\(\dfrac{b}{c}\)+ \(\dfrac{c}{a}\))>=2.3=6

Pmin khi a=b=c=1

Áp dụng bđt : \(1+b^2>=2b\)

bđt cauchy : \(\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}>3\sqrt[3]{}\) a\b . b\c . c\a = 3

\(P\ge\dfrac{3abc}{2abc}+\dfrac{a^2+b^2}{c^2+\dfrac{a^2+b^2}{2}}+\dfrac{b^2+c^2}{a^2+\dfrac{b^2+c^2}{2}}+\dfrac{c^2+a^2}{b^2+\dfrac{c^2+a^2}{2}}\)

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

Đặt \(\left(a^2+b^2;b^2+c^2;a^2+c^2\right)=\left(x;y;z\right)\)

\(\Rightarrow P\ge\dfrac{3}{2}+2\left(\dfrac{x}{y+z}+\dfrac{y}{z+x}+\dfrac{z}{x+y}\right)=\dfrac{3}{2}+2\left(\dfrac{x^2}{xy+xz}+\dfrac{y^2}{yz+xy}+\dfrac{z^2}{xz+yz}\right)\)

\(P\ge\dfrac{3}{2}+\dfrac{2\left(x+y+z\right)^2}{2\left(xy+yz+zx\right)}\ge\dfrac{3}{2}+\dfrac{3\left(xy+yz+zx\right)}{xy+yz+zx}=3+\dfrac{3}{2}=\dfrac{9}{2}\)

Dấu "=" xảy ra khi \(a=b=c\)

Ta có: \(\sqrt{a^2+b^2+c^2}\ge\sqrt{\dfrac{\left(a+b+c\right)^2}{3}}=\sqrt{3};\sqrt{a^2+b^2+c^2}\le\sqrt{\left(a+b+c\right)^2}=3\).

Đặt \(\sqrt{a^2+b^2+c^2}=t\) \((\sqrt{3}\leq t\leq 3)\).

Ta có: \(P=t+\dfrac{9-t^2}{4}+\dfrac{1}{t^2}=\dfrac{4t^3+9t^2-t^4+4}{4t^2}\).

\(\Rightarrow P-\dfrac{28}{9}=\dfrac{\left(3-t\right)\left(9t^3-9t^2+4t+12\right)}{36}\).

Do \(\sqrt{3}\le t\le3\) nên \(3-t\geq 0\); \(9t^3-9t^2+4t+12>4t+12>0\).

Nên \(P\ge\dfrac{28}{9}\).

Đẳng thức xảy ra khi t = 3, tức (a, b, c) = (0; 0; 3) và các hoán vị.

Vậy...

Lời giải:
\(P=\frac{3}{ab+bc+ac}+\frac{5}{(a+b+c)^2-2(ab+bc+ac)}=\frac{3}{ab+bc+ac}+\frac{5}{1-2(ab+bc+ac)}\)

\(=\frac{3}{x}+\frac{5}{1-2x}\) với $x=ab+bc+ac$

Theo BĐT AM-GM:
$1=(a+b+c)^2\geq 3(ab+bc+ac)$

$\Rightarrow x=ab+bc+ac\leq \frac{1}{3}$

Vậy ta cần tìm min $P=\frac{3}{x}+\frac{5}{1-2x}$ với $0< x\leq \frac{1}{3}$

Áp dụng BĐT Bunhiacopxky:

$(\frac{3}{x}+\frac{5}{1-2x})[2x+(1-2x)]\geq (\sqrt{6}+\sqrt{5})^2$

$\Leftrightarrow P\geq (\sqrt{6}+\sqrt{5})^2=11+2\sqrt{30}$

Vậy $P_{\min}=11+2\sqrt{30}$

Giá trị này đạt tại $x=3-\sqrt{\frac{15}{2}}$

\(P=\dfrac{a^2+b^2+c^2+2\left(ab+bc+ca\right)+1}{a+b+c-abc}=\dfrac{\left(a+b+c\right)^2+1}{a+b+c-abc}\ge\dfrac{\left(a+b+c\right)^2+1}{a+b+c}\)

\(\Rightarrow P\ge a+b+c+\dfrac{1}{a+b+c}\) (1)

\(P=\dfrac{a^2+b^2+c^2+3\left(ab+bc+ca\right)}{\left(a+b+c\right)\left(ab+bc+ca\right)-abc}=\dfrac{\left(a+b\right)\left(b+c\right)+\left(b+c\right)\left(c+a\right)+\left(a+b\right)\left(c+a\right)}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\)

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

\(P=\dfrac{1}{a+b+c}\left(3+\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\right)\ge\dfrac{1}{a+b+c}\left(3+\dfrac{\left(a+b+c\right)^2}{2\left(ab+bc+ca\right)}\right)\)

\(P\ge\dfrac{1}{a+b+c}\left(3+\dfrac{\left(a+b+c\right)^2}{2}\right)=\dfrac{3}{a+b+c}+\dfrac{a+b+c}{2}\)

\(\Rightarrow3P\ge\dfrac{3}{2}\left(a+b+c\right)+\dfrac{9}{a+b+c}\) (2)

Cộng vế (1) và (2):

\(\Rightarrow4P\ge\dfrac{5}{2}\left(a+b+c\right)+\dfrac{10}{a+b+c}\ge2\sqrt{\dfrac{50\left(a+b+c\right)}{2\left(a+b+c\right)}}=10\)

\(\Rightarrow P\ge\dfrac{5}{2}\)

Dấu "=" xảy ra khi \(\left(a;b;c\right)=\left(1;1;0\right)\) và các hoán vị