a+b+c=2p

\(2bc+b^2+c^2-a^2-4p\left(p-a\right)\)

\(=\left(b+c\right)^2-a^2-2p\left(2p-2a\right)\)

=(b+c-a)(b+c+a)-(a+b+c)(a+b+c-2a)

=(b+c-a)(b+c+a)-(b+c+a)(b+c-a)

=0

=>S=0

=>S-1=-1

a: VT=(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)

\(=(x^2-ax-bx+ab)+(x^2-bx-cx+bc)+(x^2-cx-ax+ca)\)

\(=3x^2-2ax-2bx-2cx+ab+bc+ca\)

\(=3x^2-2x(a+b+c)+ab+bc+ca\)

\(=3x^2-2x\cdot2x+ab+bc+ca=-x^2+ab+bc+ca\)

=VP

b: \(2bc+b^2+c^2-a^2\)

\(=\left(b+c\right)^2-a^2=\left(b+c+a\right)\left(b+c-a\right)\)

\(=2p\left(b+c+a-2a\right)=2p\left(2p-2a\right)=4p\left(p-a\right)\)

#)Giải :

Ta có : \(a+b+c=2p\)

\(\Rightarrow b+c=2p-a\)

\(\Rightarrow\left(b+c\right)^2=\left(2p-a\right)^2\)

\(\Rightarrow b^2+c^2+2bc=4p^2-4pa+a^2\)

\(\Rightarrow2bc+b^2+c^2-a^2=4p\left(p-a\right)\)

\(\Rightarrowđpcm\)

BĐT cần chứng minh tương đương:

\(a^2+b^2+c^2\ge2ab-2bc+2ca\)

\(\Leftrightarrow a^2+b^2+c^2+2bc-2a\left(b+c\right)\ge0\)

\(\Leftrightarrow a^2+\left(b+c\right)^2-2a\left(b+c\right)\ge0\)

\(\Leftrightarrow\left(a-b-c\right)^2\ge0\) (luôn đúng)

Vậy BĐT đã cho đúng

Đề là tìm GTNN hay GTLN hả bạn?

Vì a,b,c là 3 cạnh tam giác nên \(a+b>c\Leftrightarrow ac+bc>c^2\)

CMTT: \(ab+bc>b^2;ab+ac>a^2\)

Cộng vế theo vế \(\Leftrightarrow a^2+b^2+c^2< ab+bc+ca+ab+bc+ca\)

\(\Leftrightarrow a^2+b^2+c^2< 2ab+2bc+2ca\\ \Leftrightarrow a^2+b^2+c^2-2ab-2bc-2ca< 0\)

a)

$A=\dfrac{a}{b^2+1}+\dfrac{b}{c^2+1}+\dfrac{c}{a^2+1}$

$\ge \dfrac{(a+b+c)^2}{a(b^2+1)+b(c^2+1)+c(a^2+1)}\qquad (\text{Cauchy Engel})$

$=\dfrac{1}{ab^2+bc^2+ca^2+1}$

$\ge \dfrac{1}{ab(a+b)+bc(b+c)+ca(c+a)+1}$

$=\dfrac{1}{(a+b+c)(ab+bc+ca)+1}

$\ge \dfrac{1}{\frac13+1}$ $=\dfrac34$

Dấu bằng khi $a=b=c=\dfrac13$.

$\boxed{A_{\min}=\dfrac34}$

b)

$B=\dfrac{a}{ab+2c}+\dfrac{b}{bc+2a}+\dfrac{c}{ca+2b}$

$\ge \dfrac{(a+b+c)^2}{a(ab+2c)+b(bc+2a)+c(ca+2b)}$

$=\dfrac4{a^2b+b^2c+c^2a+2(ab+bc+ca)}$

Lại có $a^2b+b^2c+c^2a\le (a+b+c)(ab+bc+ca)$$=2(ab+bc+ca)$

Nên $B\ge \dfrac4{4(ab+bc+ca)}$$=\dfrac1{ab+bc+ca}$

$\ge \dfrac1{\frac{(a+b+c)^2}{3}}$ $=\dfrac34$

Dấu bằng khi $a=b=c=\dfrac23$.

$B_{\min}=\dfrac34$