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Đặt \(\frac{a-b}{c}=x;\frac{b-c}{a}=y;\frac{c-a}{b}=z\Rightarrow\frac{c}{a-b}=\frac{1}{x};\frac{a}{b-c}=\frac{1}{y};\frac{b}{c-a}=\frac{1}{z}\)
Vì a+b+c=0 => a=-b-c ; b=-c-a ; c=-a-b
a3+b3+c3=3abc
Ta có: \(\left(\frac{a-b}{c}+\frac{b-c}{a}+\frac{c-a}{b}\right)\left(\frac{c}{a-b}+\frac{a}{b-c}+\frac{b}{c-a}\right)=\left(x+y+z\right)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)=3+\frac{x+z}{y}+\frac{x+y}{z}+\frac{y+z}{x}\)
Lại có: \(\frac{x+z}{y}=\left(x+z\right)\cdot\frac{1}{y}=\left(\frac{a-b}{c}+\frac{c-a}{b}\right)\cdot\frac{a}{b-c}=\frac{ab-b^2+c^2-ac}{bc}\cdot\frac{a}{b-c}\)
\(=\frac{a\left(b-c\right)-\left(b-c\right)\left(b+c\right)}{bc}\cdot\frac{a}{b-c}=\frac{\left(a-b-c\right)\left(b-c\right)}{bc}\cdot\frac{a}{b-c}=\frac{a\left(a+a\right)}{bc}=\frac{2a^2}{bc}=\frac{2a^3}{abc}\)
Tượng tự \(\frac{x+y}{z}=\frac{2b^3}{abc};\frac{y+z}{x}=\frac{2c^3}{abc}\)
Do đó \(\left(\frac{a-b}{c}+\frac{b-c}{a}+\frac{c-a}{b}\right)\left(\frac{c}{a-b}+\frac{a}{b-c}+\frac{b}{c-a}\right)=3+\frac{2a^3+2b^3+2c^3}{abc}=3+\frac{2\left(a^3+b^3+c^3\right)}{abc}=3+\frac{2.3abc}{abc}=9\)
=>đpcm
Đặt \(\left(\frac{a-b}{c},\frac{b-c}{a},\frac{c-a}{b}\right)\rightarrow\left(x,y,z\right)\)
Khi đó:\(\left(\frac{c}{a-b},\frac{a}{b-c},\frac{b}{c-a}\right)\rightarrow\left(\frac{1}{x},\frac{1}{y},\frac{1}{z}\right)\)
Ta có:
\(P\cdot Q=\left(x+y+z\right)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)=3+\frac{y+z}{x}+\frac{z+x}{y}+\frac{x+y}{z}\)
Mặt khác:\(\frac{y+z}{x}=\left(\frac{b-c}{a}+\frac{c-a}{b}\right)\cdot\frac{c}{a-b}=\frac{b^2-bc+ac-a^2}{ab}\cdot\frac{c}{a-b}\)
\(=\frac{c\left(a-b\right)\left(c-a-b\right)}{ab\left(a-b\right)}=\frac{c\left(c-a-b\right)}{ab}=\frac{2c^2}{ab}\left(1\right)\)
Tương tự:\(\frac{x+z}{y}=\frac{2a^2}{bc}\left(2\right)\)
\(=\frac{x+y}{z}=\frac{2b^2}{ac}\left(3\right)\)
Từ ( 1 );( 2 );( 3 ) ta có:
\(P\cdot Q=3+\frac{2c^2}{ab}+\frac{2a^2}{bc}+\frac{2b^2}{ac}=3+\frac{2}{abc}\left(a^3+b^3+c^3\right)\)
Ta có:\(a+b+c=0\)
\(\Rightarrow\left(a+b\right)^3=-c^3\)
\(\Rightarrow a^3+b^3+3ab\left(a+b\right)=-c^3\)
\(\Rightarrow a^3+b^3+c^3=3abc\)
Khi đó:\(P\cdot Q=3+\frac{2}{abc}\cdot3abc=9\)
Đặt \(A=\frac{\left(a+b\right)^2}{ab}+\frac{\left(b+c\right)^2}{bc}+\frac{\left(c+a\right)^2}{ca}=\frac{a^2+2ab+b^2}{ab}+\frac{b^2+2bc+c^2}{bc}+\frac{c^2+2ac+c^2}{ca}\)
\(=\frac{a}{b}+2+\frac{b}{a}+\frac{b}{c}+2+\frac{c}{b}+\frac{c}{a}+2+\frac{a}{c}=6+a\left(\frac{1}{b}+\frac{1}{c}\right)+b\left(\frac{1}{a}+\frac{1}{c}\right)+c\left(\frac{1}{b}+\frac{1}{a}\right)\)
\(\ge6+\frac{4a}{b+c}+\frac{4b}{c+a}+\frac{4c}{a+b}\ge6+2\left(\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{a+b}\right)+2\left(\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\right)\)
\(\ge6+2\cdot\frac{3}{2}+2\left(\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\right)=9+2\left(\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\right)\)
Dấu "=" xảy ra <=> a=b=c
Ta có
\(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}=0\)
\(\Rightarrow\frac{a}{b-c}=\frac{b}{a-c}+\frac{c}{b-a}=\frac{b^2-ab+ac-c^2}{\left(a-c\right)\left(b-a\right)}\)
\(\Rightarrow\frac{a}{\left(b-c\right)^2}=\frac{b^2-ab+ac-c^2}{\left(a-c\right)\left(b-a\right)\left(b-c\right)}\)
Tương tự
\(\frac{b}{\left(c-a\right)^2}=\frac{c^2-bc+ab-a^2}{\left(a-c\right)\left(b-a\right)\left(b-c\right)}\)
\(\frac{c}{\left(a-b\right)^2}=\frac{a^2-ac+bc-b^2}{\left(a-c\right)\left(b-a\right)\left(b-c\right)}\)
\(\Rightarrow\frac{a}{\left(b-c\right)^2}+\frac{b}{\left(c-a\right)^2}+\frac{c}{\left(a-b\right)^2}=\frac{b^2-ab+ac-c^2+c^2-bc+ab-a^2+a^2-ac+bc-b^2}{\left(a-c\right)\left(b-a\right)\left(a-b\right)}\)
=0 ( ĐPCM)
Đặt A = \(\frac{a-b}{c}+\frac{b-c}{a}+\frac{c-a}{b}\)
B = \(\frac{c}{a-b}+\frac{a}{b-c}+\frac{b}{c-a}\)
\(\Rightarrow\)A . B = 9
Ta có : A = \(\frac{a-b}{c}+\frac{b-c}{a}+\frac{c-a}{b}\)
Nhân abc với A ta được:
Aabc = \(\frac{abc\left(a-b\right)}{c}+\frac{abc\left(b-c\right)}{a}+\)\(\frac{abc\left(c-a\right)}{b}\)
Aabc = ab.( a - b ) + bc.( b - c ) + ac.( c - a )
Aabc = ab.( a - b ) + bc.( a - c + b - a ) + ac.( a - c )
Aabc = ab.( a - b ) - bc.( a - b ) - bc.( c - a ) + ac.(c - a )
Aabc = b.( a - b ).( a - c ) - c.( a - b ).(c - a )
Aabc= ( a - b ).( a - c ).( b - c )
A = \(\frac{\left(a-b\right).\left(a-c\right).\left(b-c\right)}{abc}\)
Xét a + b + c = 0 \(\Rightarrow\) a + b = - c ; c + a = -b ; b + c = -a
Nhân ( a - b ).( c - a ).( b - c ) với B ta được :
B( a - b).( c - a ).( b - c ) = \(\frac{c\left(a-b\right).\left(c-a\right).\left(b-c\right)}{a-b}\)+ \(\frac{a\left(a-b\right).\left(b-c\right).\left(c-a\right)}{b-c}\)+ \(\frac{b\left(a-b\right).\left(b-c\right).\left(c-a\right)}{c-a}\)
B( a - b ).( c - a ).( b - c ) = c.( c - a ).( b - c ) + a.( b - c ).( c - a ) + b.( a - b ).( b - c)
B( a - b ).( c - a ) .( b - c ) = c.( c - a ).( b - c ) + ( a - b ).( -b - c ).( c - a ) + b.( a - b ).( b - c )
B( a - b ).( c - a ).( b - c ) = c.( c - a ).( b - c ) - b.( a - b ).( c- a ) + b.( a - b ).(b - c ) - c.( a - b ).( c - a )
B( a - b ).( c - a ).( b - c ) = c.( c - a ).( -a + 2b - c ) + b.( a - 2c +b).(a - b )
B( a - b).( c - a ).( b - c ) = -3bc.( b + c - 2a )
B( a - b ).( c - a ).( b - c ) = -9abc
B = \(\frac{9abc}{\left(a-b\right).\left(c-a\right).\left(b-c\right)}\)
NHÂN A VỚI B :
\(\frac{\left(a-b\right).\left(b-c\right).\left(a-c\right)}{abc}\)\(.\)\(\frac{9abc}{\left(a-b\right).\left(b-c\right).\left(c-a\right)}\)= 9
\(\Rightarrow\left(\frac{a-b}{c}+\frac{b-c}{a}+\frac{c-a}{b}\right).\)\(\left(\frac{c}{a-b}+\frac{a}{b-c}+\frac{b}{c-a}\right)=9\)
MÌNH CŨNG KHÔNG CHẮC LẮM !