By the AM-QM inequality, \[\frac{x_2 + x_3 + \dots + x_{101}}{100} \le \sqrt{\frac{x_2^2 + x_3^2 + \dots + x_{101}^2}{100}}.\]Then $ x_2 + x_3 + \dots + x_{101} \le 10 \sqrt{x_2^2 + x_3^2 + \dots + x_{101}^2}, $ so \[x_1 x_2 + x_1 x_3 + \dots + x_1 x_{101} \le 10x_1 \sqrt{x_2^2 + x_3^2 + \dots + x_{101}^2} = 10x_1 \sqrt{1 - x_1^2}.\]By the AM-GM inequality, \[x_1 \sqrt{1 - x_1^2} \le \frac{x_1^2 + (1 - x_1^2)}{2} = \frac{1}{2},\]so $ 10x_1 \sqrt{1 - x_1^2} \le 5 $. Equality occurs when $ x_1 = \frac{1}{\sqrt{2}} $ and $ x_2 = x_3 = \dots = x_{101} = \frac{1}{10 \sqrt{2}}, $ so the maximum value is $ \boxed{5} $.