Let x_1, ..., x_n be in [-a, b], a,b>0. Suppose x_1 + ... + x_n = 0. Prove x_1^2 + ... + x_n^2 <= nab.

[OfficialAnu]

Let [math]x_1,\, x_2,\, ...,\, x_n[/math] be real numbers in the interval [math][-a,\, b][/math], where both [math]a[/math] and [math]b[/math] are positive. Suppose that [math]x_1\, +\, x_2\, +\, ...,\, +x_n\, =\, 0.[/math] Prove that [math]x_1^2\, +\, x_2^2\, +\, ...,\, +\, x_n^2\, \leq\, nab[/math]
I don't know what to do to begin even

 

[MarkFL]

Are you familiar with cyclic symmetry?