Can this list be balanced?
Pyth, 12 10 bytes

!%ys*VQUQs


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Saved 2 bytes thanks to Mr. Xcoder and Erik the Outgolfer.

Explanation

!%ys*VQUQs
    *VQUQ    Multiply each input by its index.
  ys         Take twice the sum (to handle half-integer positions).
!%       sQ  Check if that's a multiple of the total weight.