\(a^{2000}+b^{2000}=a^{2001}+b^{2001}=a^{2002}+b^{2002}\)
\(\Rightarrow\left(a^{2000}+b^{2000}\right)\left(a^{2002}+b^{2002}\right)=\left(a^{2001}+b^{2001}\right)^2\)
\(\Rightarrow a^{4002}+b^{4002}+a^{2000}.b^{2002}+a^{2002}.b^{2000}=a^{4002}+b^{4002}+2a^{2001}b^{2001}\)
\(\Leftrightarrow a^{2000}.b^{2000}.\left(b^2+a^2-2ab\right)=0\)
\(\Leftrightarrow a^{2000}.b^{2000}\left(a-b\right)^2=0\)
\(\Rightarrow a=b\) (do a;b dương)
Thay vào pt ban đầu:
\(2a^{2000}=2a^{2001}\Leftrightarrow2a^{2001}\left(a-1\right)=0\)
\(\Rightarrow a=1\) \(\Rightarrow b=1\)
\(\Rightarrow a^{2011}+b^{2011}=2\)
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