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I have 8 variables, all between 0 and 1. I have a system of 4 inequalities as follows

$\mid a_1 - x_1 \mid - \mid b_1 - x_1 \mid + b_2 -a_2 < 0$

$\mid b_1 - y_1 \mid - \mid a_1 - y_1 \mid + a_2 -b_2 < 0$

$\mid a_1 - y_1 \mid - \mid a_1 - x_1 \mid + x_2 -y_2 < 0$

$\mid b_1 - x_1 \mid - \mid b_1 - y_1 \mid + y_2 -x_2 < 0$

I have tried to carefully design a system of values that satisfies the inequalities, and by this process I have convinced myself that such numbers do not exist. Is there a way to prve it without expanding the absolute values?

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Add all the equations together. You get $0<0$.

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  • 1
    $\begingroup$ Lovely argument! $\endgroup$ – John Hughes Jun 24 '17 at 22:33

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