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Let $A$ be a $n \times n$ matrix over field $F$. Let $a_1, \cdots, a_n$ be the column vectors of $A$. For any subset $S \subseteq [n] = \{1, 2, \cdots, n\}$, let $a_S = \sum_{i \in S} a_i$. Alon's ...

So I walked into this very innocent-looking combinatorics problem, and quite soon I ended up with the problem to prove that any doubly stochastic $n \times n$ matrix has a non-zero permanent. Now ...