thay trực tiếp giả thiết ta có

\(\sqrt{\left(a^2+1\right)}=\sqrt{a^2+ab+bc+ac}=\sqrt{a\left(a+b\right)+c\left(a+b\right)}=\sqrt{\left(a+c\right)\left(a+b\right)}\)

tương tự ta có

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

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

nên

\(\sqrt{\left(a^2+1\right)\left(b^2+1\right)\left(c^2+1\right)}=\sqrt{\left(\left(a+b\right)\left(a+c\right)\left(b+c\right)\right)^2}=\left|\left(a+b\right)\left(a+c\right)\left(b+c\right)\right|\)

mà \(a,b,c\in Q\) nên \(\left|\left(a+b\right)\left(a+c\right)\left(b+c\right)\right|\in Q\Rightarrowđpcm\)

\(\frac{ab}{a+b}=\frac{ac}{a+c}=\frac{bc}{b+c}\Rightarrow\frac{abc}{c\left(a+b\right)}=\frac{abc}{b\left(a+c\right)}=\frac{abc}{a\left(b+c\right)}\)

\(\Rightarrow c\left(a+b\right)=b\left(a+c\right)\Leftrightarrow ac+bc=ab+bc\Rightarrow ac=ab\Rightarrow c=b\) (1)

\(\Rightarrow b\left(a+c\right)=a\left(b+c\right)\Leftrightarrow ab+bc=ab+ac\Rightarrow bc=ac\Rightarrow b=a\) (2)

\(\Rightarrow c\left(a+b\right)=a\left(b+c\right)\Leftrightarrow ac+bc=ab+ac\Rightarrow bc=ab\Rightarrow c=a\) (3)

Từ (1) ; (2) ; (3) => \(a=b=c\) (ĐPCM)

gt <=>     \(a^2+b^2+c^2-ab-bc-ca=0\)

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

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

<=>    \(\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\)        (1)

TA LUÔN CÓ:     \(\left(a-b\right)^2;\left(b-c\right)^2;\left(c-a\right)^2\ge0\forall a;b;c\)

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

TỪ (1) VÀ (2) =>    DẤU "=" SẼ XẢY RA <=>     \(\left(a-b\right)^2=\left(b-c\right)^2=\left(c-a\right)^2=0\)

<=>     \(a=b=c\)

VẬY TA CÓ ĐPCM.

a² + b² + c² = ab + bc + ca

<=> 2( a² + b² + c² ) = 2( ab + bc + ca )

<=> 2a² + 2b² + 2c² - 2ab - 2bc - 2ca = 0

<=> ( a² - 2ab + b² ) + ( b² - 2bc + c² ) + ( c² - 2ca + a² ) = 0

<=> ( a - b )² + ( b - c )² + ( c - a )² = 0 (*)

Ta có : \(\hept{\begin{cases}\left(a-b\right)^2\\\left(b-c\right)^2\\\left(c-a\right)^2\end{cases}}\ge0\forall a,b,c\Rightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\ge0\)

Đẳng thức xảy ra ( tức là (*) xảy ra ) <=> \(\hept{\begin{cases}a-b=0\\b-c=0\\c-a=0\end{cases}}\Rightarrow a=b=c\)

=> ĐPCM

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

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

\(=>3ab+3bc+3ca-a^2-b^2-c^2-2ab-2bc-2ca=0\)

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

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

\(=>2a^2+2b^2+2c^2-2ab-2bc-2ca=0\)

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

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

Tổng 3 số không âm =0 <=> chúng = 0

<=> \(\hept{\begin{cases}a-b=0\\b-c=0\\c-a=0\end{cases}< =>a=b=c\left(đpcm\right)}\)

bài này trong sách nâng cao và phát triển à

Vì là trong sách nên có lẽ đã lm đc câu a nên ta sẽ áp dụng:

b) 2.( ab+bc+ca) = 2( a^2.b^2+ b^2.c^2+c^2.a^2+ 2.b^2.a.c + 2a^2.b.c+ 2c^2.a.b)

= 2. [ a^2.b^2+b^2.c^2+c^2.a^2+ 2abc ( a+b+c)]

= 2. (a^2.b^2 + b^2.c^2 + c^2.a^2 )  ( Vì a+b+c = 0)

= a^4 + b^4 + c^4 ( theo câu a nha)

Ta có :

\(\frac{ab}{a+b}=\frac{bc}{b+c}=\frac{ca}{c+a}=\frac{ab-bc}{\left(a+b\right)-\left(b+c\right)}=\frac{bc-ca}{\left(b+c\right)-\left(c+a\right)}=\frac{ab-ca}{\left(a+b\right)-\left(c+a\right)}\)

\(\Rightarrow a=b=c\)

\(\Rightarrow Q=\frac{ab^2+bc^2+ca^2}{a^3+b^3+c^3}=1\)