a) Áp dụng nhiều lần công thức \(\left(x+y\right)^3=x^3-y^3+3xy\left(x+y\right)\), ta có:

\(\left(a+b+c\right)^3-a^3-b^3-c^3\)

\(=\left[\left(a+b\right)+c\right]^3-a^3-b^3-c^3\)

\(=\left(a+b\right)^3+c^3+3c\left(a+b\right)\left(a+b+c\right)-a^3-b^3-c^3\)

\(=a^3+b^3+3ab\left(a+b\right)+c^3+3c\left(a+b\right)\left(a+b+c\right)-a^3-b^3-c^3\)

\(=3\left(a+b\right)\left(ab+ac+bc+c^2\right)\)

\(=3\left(a+b\right)\left[a\left(b+c\right)+c\left(b+c\right)\right]\)

\(=3\left(a+b\right)\left(b+c\right)\left(a+c\right)\left(Đpcm\right)\)

b) Ta có:

\(a^3+b^3+c^3-3abc\)

\(=a^3+3ab\left(a+b\right)+b^2+c^3-3abc-3ab\left(a+b\right)\)

\(=\left(a+b\right)^3+c^3-3ab\left(a+b+c\right)\)

\(=\left(a+b+c\right)\left(a^2+2ab+b^2-ac-bc+c^2\right)-3ab\left(a+b+c\right)\)

\(=\left(a+b+c\right)\left(a^2+2ab+b^2-ac-bc+c^2-3ab\right)\)

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

Mình nghĩ bằng thế này mới đúng, bạn chắc ghi sai đề rồi

a) Ta có: (a + b + c)³ - a³ - b³ - c³ = [ (a + b + c)³ - a³ ] - ( b³ + c³)

= (a + b + c - a) ( a² + b² + c² + 2ab + 2bc + 2ac + a² + ab + ac + a²) - (b + c) ( b² - bc + c³)

= (b + c) ( 3a² + b² + c² + 3ab + 2bc + 3ac) - (b + c) ( b² - bc + c³)

= ( b + c) ( 3a² + b² + c² + 3ab + 2bc + 3ac - b² + bc - c³)

= ( b + c) ( 3a² + 3ab + 3bc + 3ac)

= 3 (b + c) [a (a + b) + c (a + b)]

= 3 (b + c) (a + b) (a + c) (đpcm)

a )

`VP= (a+b)^3-3ab(a+b)`

     `=a^3+3a^2b+3ab^2+b^3-3a^2b-3ab^2`

     `=a^3+b^3 =VT (đpcm)`

b) 

b) Ta có

a) Ta có:

`VP= (a+b)^3-3ab(a+b)`

     `=a^3 + b^3+3ab ( a + b )- 3ab ( a + b )`

     `=a^3 + b^3=VT(dpcm)`

b) Ta có

a^3+b^3+c^3-3abc

=(a+b)^3+c^3-3ab(a+b)-3bca

=(a+b+c)(a^2+2ab+b^2-ac-bc+c^2)-3ab(a+b+c)

=(a+b+c)(a^2+b^2+c^2-ab-ac-bc)

A=1

chuẩn

Do \(a+b+c=1\) nên BĐT cần chứng minh tương đương:

\(2\left(a^3+b^3+c^3\right)+3abc\ge\left(ab+bc+ca\right)\left(a+b+c\right)\)

\(\Leftrightarrow2\left(a^3+b^3+c^3\right)\ge ab\left(a+b\right)+bc\left(b+c\right)+ca\left(c+a\right)\)

Thật vậy, ta có:

\(2\left(a^3+b^3+c^3\right)=\left(a^3+b^3\right)+\left(b^3+c^3\right)+\left(c^3+a^3\right)\)

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

\(\ge\left(a+b\right)\left(2ab-ab\right)+\left(b+c\right)\left(2bc-bc\right)+\left(c+a\right)\left(2ca-ca\right)\)

\(=ab\left(a+b\right)+bc\left(b+c\right)+ca\left(c+a\right)\) (đpcm)

Dấu "=" xảy ra khi \(a=b=c=\dfrac{1}{3}\)

Đặt \(P=\dfrac{a^3}{a^2+b^2+ab}+\dfrac{b^3}{b^2+c^2+bc}+\dfrac{c^3}{c^2+a^2+ca}\)

Ta có: \(\dfrac{a^3}{a^2+b^2+ab}=a-\dfrac{ab\left(a+b\right)}{a^2+b^2+ab}\ge a-\dfrac{ab\left(a+b\right)}{3\sqrt[3]{a^3b^3}}=a-\dfrac{a+b}{3}=\dfrac{2a-b}{3}\)

Tương tự: \(\dfrac{b^3}{b^2+c^2+bc}\ge\dfrac{2b-c}{3}\) ; \(\dfrac{c^3}{c^2+a^2+ca}\ge\dfrac{2c-a}{3}\)

Cộng vế:

\(P\ge\dfrac{a+b+c}{3}=673\)

Dấu "=" xảy ra khi \(a=b=c=673\)

a+b+c=1; a>0; b>0; c>0

=>a>=b>=c>=0

=>a(a-c)>=b(b-c)>=0

=>a(a-b)(a-c)>=b(a-b)(b-c)

=>a(a-b)(a-c)+b(b-a)(b-c)>=0

mà (a-c)(b-c)*c>=0 và c(c-a)(c-b)>=0 

nên a(a-b)(a-c)+b(b-a)(b-c)+(a-c)(b-c)*c>=0

=>a^3+b^3+c^3+3acb>=a^2b+a^2c+b^2c+b^2a+c^2b+c^2a

=>a^3+b^3+c^3+6abc>=(a+b+c)(ab+bc+ac)

=>a^3+b^3+c^3+6abc>=(ab+bc+ac)

mà a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-ac-bc)

nên 2(a^3+b^3+c^3)+3acb>=a^2+b^2+c^2>=ab+bc+ac(ĐPCM)