Ta có: ${{c}^{2}}={{a}^{2}}+{{b}^{2}}\ge \frac{1}{2}{{(a+b)}^{2}}\Rightarrow \frac{a+b}{c}\le \sqrt{2}$

$\Rightarrow {{B}_{1}}=500{{\log }_{2}}{{\left( \frac{a+b}{c} \right)}^{2}}\le 500{{\log }_{2}}2=500\Rightarrow {{b}_{1}}=500$

Lại có $25{{c}^{2}}=({{3}^{2}}+{{4}^{2}})({{a}^{2}}+{{b}^{2}})\ge {{(3a+4b)}^{2}}\Rightarrow 3a+4b\le 5c\Rightarrow \frac{3a+4b}{c}\le 5$

$\Rightarrow {{B}_{2}}={{\log }_{2}}{{\left( \frac{3a+4b}{c} \right)}^{1000}}\le {{\log }_{2}}{{5}^{1000}}=1000{{\log }_{2}}5\Rightarrow {{b}_{2}}=1000{{\log }_{2}}5$