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\(\Leftrightarrow2\left(a^{2010}+b^{2010}+c^{2010}\right)=2\left(a^{1005}b^{1005}+b^{1005}c^{1005}+c^{1005}a^{1005}\right)\)
\(\Leftrightarrow2a^{2010}+2b^{2010}+2c^{2010}-2a^{1005}b^{1005}-2b^{1005}c^{1005}-2c^{1005}a^{1005}=0\)
\(\Leftrightarrow\left(a^{2010}-2a^{1005}b^{1005}+b^{2010}\right)+\left(b^{2010}-2b^{1005}c^{1005}+c^{2010}\right)+\left(c^{2010}-2c^{1005}a^{1005}+a^{2010}\right)=0\)
\(\Leftrightarrow\left(a^{1005}-b^{1005}\right)^2+\left(b^{1005}-c^{1005}\right)^2+\left(c^{1005}-a^{1005}\right)^2=0\)
\(\Rightarrow\left(a^{1005}-b^{1005}\right)^2=0;\left(b^{1005}-c^{1005}\right)^2=0;\left(c^{1005}-a^{1005}\right)^2=0\)
\(\Rightarrow a=b=c\)
\(\Rightarrow\left(a-a\right)^{20}+\left(a-a\right)^{11}+\left(a-a\right)^{2010}=0\)
2 ( a trên 2010 + b trân 2010 + c trên 2010 ) = 2 ( a trên 1005 b trên 1005 + b trên 1005 c trên 1005 + c trên 1005 a trên 1005 )
2a^ ( 2010 ) + 2b^ ( 2010 ) + 2c^ ( 2010 ) - 2a^ ( 1005 ) b^ ( 1005 ) - 2b^ ( 1005 ) c^ ( 1005 ) - 2c^ ( 1005 )a^ ( 1005 ) = O\)
( a^ ( 2010 ) - 2a^ ( 1005 ) b^ ( 1005 ) + b^ ( 2010 ) + ( b^( 2010 ) - 2b^ ( 1005 ) c^ ( 1005 ) + c^ ( 2010 ) + ( c^ ( 2010 ) - 2c^ ( 1005 ) a^ ( 1005 ) + a^ ( 2010 ) = 0\)
( a^ ( 1005 ) ^2 + ( b^ ( 1005 ) - c^ ( 1005 ) ^2 + ( c^ ( 1005 ) - a^ ( 1005 ) - a^ ( 1005 ) ^2 = 0\)
( a^ ( 1005 ) - b^ ( 1005 ) ^ 2= 0 : ( b^ ( 1005 ) - c^ ( 1005 ) ^2 = 0 : ( c^ ( 1005 ) - a^ ( 1005 ) ^2 = 0\)
a = b = c
( a - a ) ^ ( 20 ) + ( a - a ) ^ ( 11 ) + ( a - a ) ^ (2010 = 0\)
Vậy : ( a -a ) ^ ( 20 ) + ( a - a ) ^ ( 11 ) + ( a + a ) ^ ( 2010 = 0\)
\(a^{2010}+b^{2010}+c^{2010}=a^{1005}b^{1005}+b^{1005}c^{1005}+a^{1005}c^{1005}\)
=>\(2a^{2010}+2b^{2010}+2c^{2010}-2a^{1005}b^{1005}-2b^{1005}c^{1005}-2a^{1005}c^{1005=0}\)
=>\(\left(a^{1005}-b^{1005}\right)\left(b^{1005}-c^{1005}\right)\left(a^{1005}-c^{1005}\right)=0\)
=>a=b=c
\(A=\left(b-b\right)^{20}+\left(b-b\right)^{11}+\left(c-c\right)^{2010}=0\)
1/a+b+c=0
\(\Rightarrow a+c=-b\)
\(\Rightarrow a=-b-c\)
2/\(a^2+b^2+c^2=2010\)
\(\Rightarrow a^2+c^2=2010-b^2\)
\(\Rightarrow a^2=2010-b^2-c^2\)
\(\Rightarrow a=\pm2010-b^2-c^2\)
ta có : \(a^2+b^2+c^2=ab+bc+ca\Leftrightarrow2a^2+2b^2+2c^2=2ab+2bc+2ca\)
\(\Leftrightarrow a^2-2ab+b^2+b^2-2bc+c^2+c^2-2ca+a^2=0\)
\(\Leftrightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\) \(\Leftrightarrow a=b=c\)
\(\Rightarrow C=\dfrac{a^{2010}+b^{2010}}{c^{2010}}+\dfrac{b^{2010}+c^{2010}}{a^{2010}}+\dfrac{c^{2010}+a^{2010}}{b^{2010}}=3\dfrac{a^{2010}+a^{2010}}{a^{2010}}\)
\(=3\dfrac{2a^{2010}}{a^{2010}}=3.2=6\)