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Do \(a,b,c\) là các số dương suy ra:
\(a>0;b>0;c>0\)
Suy ra: \(a+b+c>0\)
Ta có: \(a^3+b^3+c^3=3abc\)
\(\Leftrightarrow\left(a+b\right)^3-3ab\left(a+b\right)+c^3-3abc=0\)
\(\Leftrightarrow\left(a+b+c\right)\left[\left(a+b\right)^2-\left(a+b\right)c+c^2-3ab\left(a+b+c\right)\right]=0\)
\(\Leftrightarrow\left(a+b+c\right)\left(a^2+2ab+b^2-ac-bc+c^2-3ab\right)=0\)
\(\Leftrightarrow\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ca\right)=0\)
\(\Rightarrow a+b+c=0\) hoặc \(a^2+b^2+c^2-ab-bc-ca=0\)
Do \(a+b+c>0\)
Suy ra: \(a^2+b^2+c^2-ab-bc-ca=0\)
\(2a^2+2b^2+2c^2-2ab-2bc-2ca=0\)
\(\left(a^2-2ab+b^2\right)+\left(b^2-2bc+c^2\right)+\left(c^2-2ca+a^2\right)=0\)
\(\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\)
Suy ra: \(a-b=0;b-c=0\) và \(c-a=0\)
Suy ra: \(a=b=c\)
Suy ra: \(\frac{a}{b}=\frac{b}{c}=\frac{c}{a}=1\)
Ta có: \(\left(\frac{a}{b}-1\right)+\left(\frac{b}{c}-1\right)+\left(\frac{c}{a}-1\right)=\left(1-1\right)+\left(1-1\right)+\left(1-1\right)=0\)
Vậy ...
Sau khi giải bài này xong mình cảm thấy hoa mắt và chóng mặt, mong GP sẽ gấp đôi :)
![](https://rs.olm.vn/images/avt/0.png?1311)
Bài làm
Ta có : a3 + b3 + c3 = 3abc
<=> ( 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 + 2ab + b2 - ac - bc + c2 - 3ab ) = 0
<=> ( a + b + c )( a2 + b2 + c2 - ab - bc - ac ) = 0
<=> \(\orbr{\begin{cases}a+b+c=0\\a^2+b^2+c^2-ab-bc-ac=0\end{cases}}\)
Vì a, b, c dương => a + b + c > 0 => a + b + c = 0 vô lí
Xét 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 . Đẳng thức xảy ra <=> \(\hept{\begin{cases}a-b=0\\b-c=0\\a-c=0\end{cases}}\Leftrightarrow\hept{\begin{cases}a=b\\b=c\\a=c\end{cases}}\Leftrightarrow a=b=c\)
=> \(P=\left(\frac{a}{b}-1\right)+\left(\frac{b}{c}-1\right)+\left(\frac{c}{a}-1\right)=\left(\frac{a}{a}-1\right)+\left(\frac{b}{b}-1\right)+\left(\frac{c}{c}-1\right)\)
\(=\left(1-1\right)+\left(1-1\right)+\left(1-1\right)\)
\(=0\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Thay a3+b3=(a+b)3-3ab(a+b) vào giả thiết ta có:
(a+b)3-3ab(a+b)+c3-3abc=0
<=> [(a+b)+c].\(\left[\left(a+b\right)^2-\left(a+b\right)c+c^2\right]\)-3ab(a+b+c)=0
<=> (a+b+c) (a2+b2+c2-ab-bc+c2-3ab)=0
<=> (a+b+c)(a2+b2+c2-ab-bc-ca)=0
\(\Leftrightarrow\orbr{\begin{cases}a+b+c=0\\a^2+b^2+c^2-ab-bc-ca=0\end{cases}}\)
- Nếu a+b+c=0
\(\Rightarrow A=\frac{b+a}{b}\cdot\frac{c+b}{c}\cdot\frac{a+c}{a}=\frac{-c}{b}\cdot\frac{-a}{c}\cdot\frac{-b}{a}\Rightarrow A=-1\)
- Nếu \(a^2+b^2+c^2-ab-bc-ca=0\)
\(\Leftrightarrow2a^2+2b^2+2c^2-2ab-2bc-2ca=0\)
\(\Leftrightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\)
<=> a=b=c
Khi đó \(A=\left(1+\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right)=\left(1+1\right)\left(1+1\right)\left(1+1\right)=8\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Sửa đề: tính P=(1+a/b)(1+b/c)(1+c/a)
\(a^3+b^3+c^3=3abc\)
\(\Leftrightarrow a^3+b^3+c^3-3abc=0\)
\(\Leftrightarrow\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ca\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}a+b+c=0\left(1\right)\\a^2+b^2+c^2-ab-bc-ca=0\left(2\right)\end{cases}}\)
- Xét (1) ta có: \(a+b+c=0\Leftrightarrow\hept{\begin{cases}-a=b+c\\-b=c+a\\-c=a+b\end{cases}}\)
=> \(P=\frac{a+b}{b}\cdot\frac{b+c}{c}\cdot\frac{c+a}{a}=\frac{\left(-c\right).\left(-a\right).\left(-b\right)}{bca}=-\frac{abc}{abc}=-1\)
- Xét (2) ta có: \(a^2+b^2+c^2-ab-bc-ca=0\)
\(\Leftrightarrow2a^2+2b^2+2c^2-2ab-2bc-2ca=0\)
\(\Leftrightarrow\left(a^2-2ab+b^2\right)+\left(b^2-2bc+c^2\right)+\left(c^2-2ca+a^2\right)=0\)
\(\Leftrightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\)
Mà \(\hept{\begin{cases}\left(a-b\right)^2\ge0\\\left(b-c\right)^2\ge0\\\left(c-a\right)^2\ge0\end{cases}\Rightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\ge0}\)
\(\Rightarrow\hept{\begin{cases}a-b=0\\b-c=0\\c-a=0\end{cases}\Rightarrow a=b=c}\)
=>\(P=\frac{a+b}{b}\cdot\frac{b+c}{c}\cdot\frac{c+a}{a}=\frac{2a}{a}\cdot\frac{2a}{a}\cdot\frac{2a}{a}=2.2.2=8\)
Vậy P=-1 hoặc P=8
Ta có; \(a^3+b^3+c^3=3abc\) hay \(a^3+b^3+c^3-3abc=0\)
Suy ra \(a+b+c=0\) hoặc a = b = c. (bạn tự chứng minh)
* Nếu a + b + c = 0 thì:
\(P=\frac{a+b}{b}.\frac{b+c}{c}.\frac{c+a}{a}=\frac{-c}{b}.\frac{-a}{c}.\frac{-b}{a}=-1\)
*Nếu a = b = c thì \(P=\left(1+1\right)\left(1+1\right)\left(1+1\right)=8\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có : \(a^3+b^3+c^3=3abc\)
\(\Rightarrow\left(a+b\right)^3+c^3-3ab\left(a+b\right)-3abc=0\)
\(\Leftrightarrow\left(a+b+c\right)\left[\left(a+b\right)^2-\left(a+b\right)c+c^2\right]-3ab\left(a+b+c\right)=0\)
\(\Leftrightarrow\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ca\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}a+b+c=0\left(1\right)\\a^2+b^2+c^2-ab-bc-ca=0\left(2\right)\end{cases}}\)
Từ (1) \(\Rightarrow\hept{\begin{cases}a+b=-c\\b+c=-a\\c+a=-b\end{cases}}\)
Khi đo s: \(P=\frac{abc}{\left(-a\right)\left(-b\right)\left(-c\right)}=-1\)
Từ (2) \(\Rightarrow\left(a^2-2ab+b^2\right)+\left(b^2-2bc+c^2\right)+\left(c^2-2ca+a^2\right)=0\)
\(\Leftrightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\)
\(\Leftrightarrow a=b=c\)
Khi đó : \(P=\frac{a^3}{2a\cdot2a\cdot2a}=\frac{1}{8}\)
Vậy : \(P=\frac{1}{8}\) hoặc \(P=-1\) với a,b,c thỏa mãn đề.
![](https://rs.olm.vn/images/avt/0.png?1311)
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 \(a^2+b^2+c^2-ab-bc-ca=0\)
\(\Leftrightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\)
\(\Rightarrow a=b=c\)
Khi đó \(A=2^3=8\)
Nếu \(a+b+c=0\Rightarrow a+b=-c;b+c=-a;c+a=-b\)
Thay vào ta được:
\(A=\frac{a+b}{b}\cdot\frac{b+c}{c}\cdot\frac{c+a}{a}=\frac{-abc}{abc}=-1\)
Vậy A = 8 hoặc A = -1