Question

cho các số dương a và b thõa mãn điều kiện a¹⁰⁰+b¹⁰⁰=a¹⁰¹+b¹⁰¹=a¹⁰²+b¹⁰² .Cmr : a+b/ab=a²+b²/ab

Answer 1

\(a^{100}+b^{100}=a^{101}+b^{101}=a^{102}+b^{102}\)

\(\Rightarrow\left(a^{100}+b^{100}\right)\left(a^{102}+b^{102}\right)=\left(a^{101}+b^{101}\right)^2\)

\(\Rightarrow a^{202}+b^{202}+a^{100}b^{102}+a^{102}b^{100}=a^{202}+b^{202}+2a^{101}b^{101}\)

\(\Rightarrow a^{100}b^{100}\left(a^2+b^2\right)=a^{100}b^{100}\left(2ab\right)\)

\(\Rightarrow a^2+b^2=2ab\)

\(\Rightarrow\left(a-b\right)^2=0\)

\(\Rightarrow a=b\)

Thế vào \(a^{100}+b^{100}=a^{101}+b^{101}\)

\(\Rightarrow a^{100}+a^{100}=a^{101}+a^{101}\)

\(\Rightarrow2a^{100}\left(a-1\right)=0\)

\(\Rightarrow a=1\Rightarrow b=1\)

\(\Rightarrow...\)