Let $x_1,$ $x_2,$ $\dots,$ $x_{101}$ be positive real numbers such that $x_1^2 + x_2^2 + \dots + x_{101}^2 = 1.$  Find the maximum value of
\[x_1 x_2 + x_1 x_3 + \dots + x_1 x_{101}.\]
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $x_1,$ $x_2,$ $\dots,$ $x_{101}$ be positive real numbers such that $x_1^2 + x_2^2 + \dots + x_{101}^2 = 1.$  Find the maximum value of
\[x_1 x_2 + x_1 x_3 + \dots + x_1 x_{101}.\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
5
Let $x_1,$ $x_2,$ $\dots,$ $x_{101}$ be positive real numbers such that $x_1^2 + x_2^2 + \dots + x_{101}^2 = 1.$  Find the maximum value of
\[x_1 x_2 + x_1 x_3 + \dots + x_1 x_{101}.\]
train