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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.

:)