Đáp án:

$\begin{array}{l}
a)\left\{ \begin{array}{l}
a + b = 9\\
ab = 20
\end{array} \right.\\
Do:a + b = 9\\
 \Rightarrow a = 9 - b\\
 \Rightarrow \left( {9 - b} \right).b = 20\\
 \Rightarrow 9b - {b^2} = 20\\
 \Rightarrow {b^2} - 9b + 20 = 0\\
 \Rightarrow {b^2} - 4b - 5b + 20 = 0\\
 \Rightarrow \left( {b - 4} \right)\left( {b - 5} \right) = 0\\
 \Rightarrow \left[ \begin{array}{l}
b = 4 \Rightarrow a = 5\\
b = 5 \Rightarrow a = 4
\end{array} \right.\\
Do:a < b\\
 \Rightarrow a = 4;b = 5\\
 \Rightarrow {\left( {a - b} \right)^{2015}} = {\left( {4 - 5} \right)^{2015}} = {\left( { - 1} \right)^{2015}} =  - 1\\
b)\\
C = {1^2} - {2^2} + {3^2} - {4^2} + {5^2} - {6^2} + .. + \\
{2013^2} - {2014^2} + {2015^2}\\
 = {1^2} + \left( {{3^2} - {2^2}} \right) + \left( {{5^2} - {4^2}} \right) + ... + \left( {{{2015}^2} - {{2014}^2}} \right)\\
 = 1 + \left( {3 - 2} \right)\left( {3 + 2} \right) + \left( {5 - 4} \right)\left( {5 + 4} \right) + ... + \\
\left( {2015 - 2014} \right)\left( {2015 + 2014} \right)\\
 = 1 + 2 + 3 + 4 + 5 + ... + 2014 + 2015\\
 = \dfrac{{\left( {2015 + 1} \right).2015}}{2}\\
 = 2015.1008\\
 = 2031120
\end{array}$