cho (a+b+c)^3=8/9+(a+b/3-c/3)^3+(b+c/3-a/3)^3+(c+a/3-b/3)^3.tính p=(a+2b)(b+2c)(c+2a)
vói a,b,c là các số thực dương
GIÚP MK NHA MN,MAI MK PHẢI NỘP RÙI
[tex]đắt:..a+\frac{b}{3}-\frac{c}{3}=x\\\\ b+\frac{c}{3}-\frac{a}{3}=y\\\\ c+\frac{a}{3}-\frac{b}{3}=z\\\\ => x+y+z=a+b+c[/tex]
=> pt <=> [tex](x+y+z)^3=\frac{8}{9}+x^3+y^3+z^3\\\\ <=> x^3+y^3+z^3+3x^2y+3xy^2+3yz^2+3y^2z+3xz^2+3x^2z+6xyz=\frac{8}{9}+x^3+y^3+z^3\\\\ <=> x^2y+xy^2+yz^2+y^2z+xz^2+x^2z+2xyz=\frac{8}{27}\\\\ <=> (x^2y+xy^2)+(yz^2+xz^2)+(y^2z+xyz)+(x^2z+xyz)=\frac{8}{27}\\\\ <=> xy.(x+y)+z^2.(x+y)+yz.(x+y)+xz.(x+y)=\frac{8}{27}\\\\ <=> (x+y).(xy+z^2+yz+xz)=\frac{8}{27}\\\\ <=> (x+y).(y+z).(z+x)=\frac{8}{27}\\\\ <=> (a+\frac{b}{3}-\frac{c}{3}+b+\frac{c}{3}-\frac{a}{3}).(b+\frac{c}{3}-\frac{a}{3}+c+\frac{a}{3}-\frac{b}{3}).(c+\frac{a}{3}-\frac{b}{3}+a+\frac{b}{3}-\frac{c}{3})=\frac{8}{27}\\\\ <=> (\frac{2a+4b}{3}).(\frac{2b+4c}{3}).(\frac{2c+4a}{3})=\frac{8}{27}\\\\ <=> (a+2b).(b+2c).(c+2a)=1 => P=1[/tex]
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