jmc

By the AM-QM inequality, $$\frac{x_2+x_3+\cdots +x_{101}}{100}\le \sqrt{\frac{x_2^2+x_3^2+\cdots +x_{101}^2}{100}}.$$Then $x_2+x_3+\cdots +x_{101}\le 10\sqrt{x_2^2+x_3^2+\cdots +x_{101}^2},$ so $$x_1{x}_2+x_1{x}_3+\cdots +x_1{x}_{101}\le 10x_1\sqrt{x_2^2+x_3^2+\cdots +x_{101}^2}=10x_1\sqrt{1-x_1^2}.$$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 $x_2=x_3=\cdots =x_{101}=\frac{1}{10\sqrt{2}},$ so the maximum value is $\boxed{5}.$