b: (3x-2)^5+(5-x)^5+(-2x-3)^5=0

Đặt a=3x-2; b=-2x-3

Pt sẽ trở thành:

a^5+b^5-(a+b)^5=0

=>a^5+b^5-(a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5)=0

=>-5a^4b-10a^3b^2-10a^2b^3-5ab^4=0

=>-5a^4b-5ab^4-10a^3b^2-10a^2b^3=0

=>-5ab(a^3+b^3)-10a^2b^2(a+b)=0

=>-5ab(a+b)(a^2-ab+b^2)-10a^2b^2(a+b)=0

=>-5ab(a+b)(a^2-ab+b^2+2ab)=0

=>-5ab(a+b)(a^2+b^2+ab)=0

=>ab(a+b)=0

=>(3x-2)(-2x-3)(5-x)=0

=>\(x\in\left\{\dfrac{2}{3};-\dfrac{3}{2};5\right\}\)

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

\(\Leftrightarrow a^2+b^2+c^2-ab-bc-ac=0\Leftrightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\)

=> a=b=c

b, \(0=\left(a+b+c\right)^3=a^3+b^3+c^3+6abc+3a^2b+3ab^2+3b^2c+3bc^2+3c^2a+3ca^2\)

\(=a^3+b^3+c^3+6abc+3ab\left(a+b\right)+3bc\left(b+c\right)+3ac\left(a+c\right)\)

\(=a^3+b^3+c^3+6abc-3abc-3abc-3abc\)

\(\Rightarrow a^3+b^3+c^3=3abc\)

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

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

\(=\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ac\right)=0\)

Từ (a) -> hoặc a+b+c = 0 hoặc a=b=c. Vậy ko thể khẳng định như vây

a³ + b³ + c³ = 3abc

⇒ a³ + b³ + c³ - 3abc = 0

⇒ ( a³ + b³ ) + c³ - 3abc = 0

⇒ ( a + b )³ - 3ab( a + b ) + c³ - 3abc = 0

⇒ [ ( a + b )³ + c³ ] - [ 3ab( a + b ) + 3abc ] = 0

⇒ ( a + b + c )[ ( a + b )² - ( a + b ).c + c² ] - 3ab( a + b + c ) = 0

⇒ ( a + b + c )( a² + b² + c² - ab - bc - ac ) = 0

⇒ \(\orbr{\begin{cases}a+b+c=0\\a^2+b^2+c^2-ab-bc-ac=0\end{cases}}\)

+) a² + b² + c² - ab - bc - ac = 0

⇒ 2( a² + b² + c² - ab - bc - ac ) = 2.0

⇒ 2a² + 2b² + 2c² - 2ab - 2bc - 2ac = 0

⇒ ( a² - 2ab + b² ) + ( b² - 2bc + c² ) + ( a² - 2ac + c² ) = 0

⇒ ( a - b )² + ( b - c )² + ( a - c )² = 0

VT ≥ 0 ∀ a,b,c . Dấu "=" xảy ra khi a = b = c

⇒ a + b + c = 0 hoặc a = b = c ( đpcm )

\(a,\dfrac{3}{a+b}=\dfrac{2}{b+c}=\dfrac{1}{c+a}\\ \Rightarrow\dfrac{a+b}{3}=\dfrac{b+c}{2}=\dfrac{c+a}{1}=\dfrac{2\left(a+b+c\right)}{6}=\dfrac{a+b+c}{3}\\ \Rightarrow\dfrac{a+b}{3}=\dfrac{a+b+c}{3}\\ \Rightarrow3\left(a+b+c\right)=3\left(a+b\right)\\ \Rightarrow3\left(a+b\right)+3c=3\left(a+b\right)\\ \Rightarrow3c=0\\ \Rightarrow c=0\)

Vậy \(P=\dfrac{a+b-2019c}{a+b+2018c}=\dfrac{a+b}{a+b}=1\)

Từ a+b+c=0 => b+c=-a 

Theo đề ra ta có a³ + b³ + c³ = 0 

=> a³ + (b+c)(b² - bc + c² )=0 

<=> a³- a[(b + c )² -3bc]= 0 

<=> a³- [( -a )² - 3bc] = 0 

<=> a³ -  a³ +3bc = 0 

<=> 3bc= 0 

<=> a =0 hoặc b=0 hoặc c=0 ( đpcm) 

cho mik điểm nha bạn ơiii

\(a+b+c=0\Rightarrow\left\{{}\begin{matrix}a+b=-c\\c+a=-b\\b+c=-a\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}A=a.\left(-c\right).\left(-b\right)=abc\\B=b.\left(-a\right).\left(-c\right)=abc\\C=c.\left(-b\right).\left(-a\right)=abc\end{matrix}\right.\)

\(\Rightarrow A=B=C\)

Ta có: \(a^3+b^3+c^3=3abc\)

\(\Rightarrow a^3+b^3+c^3-3abc=0\)

\(\Leftrightarrow\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ca\right)=0\)

\(\Leftrightarrow\orbr{\begin{cases}a+b+c=0\\a^2+b^2+c^2-ab-bc-ca=0\end{cases}}\)

\(\Leftrightarrow\orbr{\begin{cases}a+b+c=0\\2a^2+2b^2+2c^2-2ab-2bc-2ca=0\end{cases}}\)

\(\Leftrightarrow\orbr{\begin{cases}a+b+c=0\\\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\end{cases}}\)

\(\Leftrightarrow\orbr{\begin{cases}a+b+c=0\\a=b=c\end{cases}}\)

bạn thay vào M giải tiếp nha

Ta có: \(a^3+b^3+c^3=3abc\)

\(\Leftrightarrow\left(a^3+b^3\right)+c^3-3abc=0\)

\(\Leftrightarrow\left(a+b\right)^3-3ab\left(a+b\right)+c^3-3abc=0\)

\(\Leftrightarrow\left[\left(a+b\right)^3+c^3\right]-\left[3ab\left(a+b\right)+3abc\right]=0\)

\(\Leftrightarrow\left(a+b+c\right)\left[\left(a+b\right)^2-\left(a+b\right)c+c^2\right]-3ab\left(a+b+c\right)=0\)

\(\Leftrightarrow\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ca\right)=0\)

Nếu \(a^2+b^2+c^2-ab-bc-ca\)

\(=\frac{1}{2}\left[\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\right]\ge0\left(\forall a,b,c\right)\)

Dấu "=" xảy ra khi: a = b = c

Khi đó: \(M=\left(1+\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right)=\left(1+1\right)^3=8\)

Nếu \(a+b+c=0\Rightarrow\hept{\begin{cases}a+b=-c\\b+c=-a\\c+a=-b\end{cases}}\)

\(\Rightarrow M=\frac{\left(a+b\right)\left(b+c\right)\left(c+a\right)}{abc}=\frac{-abc}{abc}=-1\)

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

\(\Leftrightarrow a^3+b^3+c^3-3abc=0\)

\(\Leftrightarrow\left(a+b\right)^3-3ab\left(a+b\right)+c^3-3abc=0\)

\(\Leftrightarrow\left(a+b+c\right)\left[\left(a+b\right)^2-\left(a+b\right)c+c^2\right]-3ab\left(a+b+c\right)=0\)

\(\Leftrightarrow\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ca\right)=0\)

\(\Leftrightarrow\orbr{\begin{cases}a+b+c=0\\a^2+b^2+c^2-ab-bc-ca=0\end{cases}}\)

- \(a+b+c=0\): 

\(M=\left(1+\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right)\)

\(=\frac{\left(a+b\right)\left(b+c\right)\left(c+a\right)}{abc}\)

\(=\frac{\left(-c\right)\left(-a\right)\left(-b\right)}{abc}=-1\)

- \(a^2+b^2+c^2-ab-bc-ca=0\)

\(\Leftrightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\)

\(\Leftrightarrow a=b=c\).

\(M=\left(1+\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right)\)

\(=\left(1+1\right)\left(1+1\right)\left(1+1\right)=8\)