$2\sum x_ix_j = [\sum x_i]^2-\sum x_i^2$

But $\sum x_i^2 = 101$ [since, $x_i^2$ is always 1]

Therefore $\sum x_ix_j = \frac{[\sum x_i]^2-101}{2}$

We want the smallest positive value then $\sum x_i \geq 11$

therefore smallest value of $\sum x_i$ is $11$ then

$\sum x_ix_j =\frac{121-101}{2}=10$