Áp dụng bđt AM-GM :(a+b)($\frac{x^{4}}{a}$+$\frac{y^{4}}{b}$)$\geq$$(x^{2}+y^{2})^{2}$$\Rightarrow$$\frac{x^{4}}{a}$+$\frac{y^{4}}{b}$$\geq$$\frac{1}{a+b}$
Dấu ''='' xra $\Leftrightarrow$$\frac{x^{2}}{a}$=$\frac{y^{2}}{b}$
ta có $\frac{x^{2}}{a}$=$\frac{y^{2}}{b}$=$\frac{1}{a+b}$
$\Rightarrow$$(\frac{x^{2}}{a})^{1006}$+$(\frac{y^{2}}{b})^{1006}$=$\frac{2}{(a+b)^{1006}}$
$\Rightarrow$đpcm