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Let me state two problems that look very much alike. The first one can be solved putting together answers that different people have given to some questions I asked here a few weeks ago. The second ...

Conjecture: Let $e_1 = (1,0,\ldots,0), \ldots , e_{m_1+m_2} = (0,\ldots,0,1)$ be the unit vectors of the standard basis $E$ of $\mathbb{R}^{m_1+m_2}$. An enumeration $ E \cup -E = \{f_1, \ldots, ...