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Oct 26, 2012 at 12:56 comment added Federico Poloni The OP was interested in vectors with entries in $[0,1]$. Nevertheless, it is easy to replace your quadratic with one with solutions $1$ and $\alpha<1$, and the counterexample does not change.
Oct 26, 2012 at 11:36 history answered Klaus Draeger CC BY-SA 3.0