1. Given $S=1^{2}-2^{2}+3^{2}-4^{2}+\cdots-100^{2}+$ $101^{2}$. Then the remainder when $S$ is divided by 103 is
$=.1 .1$.
Notice that
$$
\begin{array}{l}
S=1+\left(3^{2}-2^{2}\right)+\left(5^{2}-4^{2}\right)+\cdots+\left(101^{2}-100^{2}\right) \\
=1+2+3+\cdots+100+101 \\
=\frac{101 \times 102}{2}=5151=50 \times 103+1 .
\end{array}
$$

Therefore, the required remainder is 1.