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 \(x_1 x_2 + x_1 x_3 + \dots + x_1 x_{101}.\)

Maximum occurs at \(x_1 = x_2 = x_3 = \cdots = x_{101} = \dfrac{1}{\sqrt{101}}\).

Maximum value is \(100\left(\dfrac{1}{\sqrt{101}}\right)^2 =\dfrac{100}{101}\).

Notes: The equation \(x_1^2 + x_2^2 + \cdots + x_{101}^2 = 1\) represents a 101-sphere with radius 1.

:)

I believe your answer but prove it.