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# help

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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 $$x_1 x_2 + x_1 x_3 + \dots + x_1 x_{101}.$$

Mar 8, 2019

#1
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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.

:)

Mar 9, 2019
#2
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I believe your answer but prove it.

Rom  Mar 9, 2019