Question

Cho a²+b²+c²=ab+bc+ca.Tính C=\(\dfrac{a^{2010}+b^{2010}}{c^{2010}}+\dfrac{b^{2010}+c^{2010}}{a^{2010}}+\dfrac{c^{2010}+a^{2010}}{b^{2010}}\)

Answer 1

ta có : \(a^2+b^2+c^2=ab+bc+ca\Leftrightarrow2a^2+2b^2+2c^2=2ab+2bc+2ca\)

\(\Leftrightarrow a^2-2ab+b^2+b^2-2bc+c^2+c^2-2ca+a^2=0\)

\(\Leftrightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\) \(\Leftrightarrow a=b=c\)

\(\Rightarrow C=\dfrac{a^{2010}+b^{2010}}{c^{2010}}+\dfrac{b^{2010}+c^{2010}}{a^{2010}}+\dfrac{c^{2010}+a^{2010}}{b^{2010}}=3\dfrac{a^{2010}+a^{2010}}{a^{2010}}\)

\(=3\dfrac{2a^{2010}}{a^{2010}}=3.2=6\)