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Found problems: 1

1981 Putnam, B4
Tags: matrix , rank , rank of matrix
Let $V$ be a set of $5\times7$ matrices, with real entries and closed under addition and scalar multiplication. Prove or disprove the following assertion: If $V$ contains matrices of ranks $0, 1, 2, 4,$ and $5$, then it also contains a matrix of rank $3$.