\(a^{102}+b^{102}=\left(a^{101}+b^{101}\right)\left(a+b\right)-ab\left(a^{100}+b^{100}\right)\)
Mà \(a^{100}+b^{100}=a^{101}+b^{101}=a^{102}+b^{102}\)
Khi đó:
\(a^{102}+b^{102}=\left(a^{102}+b^{102}\right)\left(a+b-ab\right)\)
\(\Rightarrow a+b-ab=1\)
\(\Rightarrow\left(a-1\right)\left(b-1\right)=0\)
\(\Rightarrow a=1;b=1\)
\(\Rightarrow a^{2010}+b^{2010}=2\)