Question

[Tony Nguyễn]

Cho a,b thuộc R; a,b dương sao cho a²⁰⁰⁹+b²⁰⁰⁹=a²⁰¹⁰+b²⁰¹⁰=a²⁰¹¹⁺b²⁰¹¹

Tính a²⁰¹²+b²⁰¹²

Answer 1

[Mr Lazy]

Đề \(\Rightarrow\left(a^{2011}+b^{2011}\right)-2\left(a^{2010}+b^{2010}\right)+\left(a^{2009}+b^{2009}\right)=0\)

\(\Leftrightarrow a^{2011}-2a^{2010}+a^{2009}+b^{2011}-2b^{2010}+b^{2009}=0\)

\(\Leftrightarrow a^{2009}\left(a^2-2a+1\right)+b^{2009}\left(b^2-2b+1\right)=0\)

\(\Leftrightarrow a^{2009}\left(a-1\right)^2+b^{2009}\left(b-1\right)^2=0\)

\(\Leftrightarrow a-1=b-1=0\text{ (do }a,\text{ }b>0\text{)}\)

\(\Leftrightarrow a=b=1\)

\(\Rightarrow a^{2012}+b^{2012}=1+1=2\)