Y by Adventure10, Mango247
Let
be
real numbers satisfying
and
. Prove that for any
vectors
in the plane, there exists a permutation
of the numbers
such that ![\[ \left |\sum_{i=1}^nx_{k_i}\vec{\alpha_i}\right | \geq \dfrac{AB}{2A+B}\max_{1\leq i\leq n}|\alpha_i|.\]](//latex.artofproblemsolving.com/b/7/8/b782ff3d60196f42d88ef81fd27c23bbaf5c8656.png)








![\[ \left |\sum_{i=1}^nx_{k_i}\vec{\alpha_i}\right | \geq \dfrac{AB}{2A+B}\max_{1\leq i\leq n}|\alpha_i|.\]](http://latex.artofproblemsolving.com/b/7/8/b782ff3d60196f42d88ef81fd27c23bbaf5c8656.png)
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