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A set of 2-by-2 matrices

Source: Putnam 1990 B3

July 12, 2013
linear algebramatrixpigeonhole principlePutnamcollege contests

Problem Statement

Let SS be a set of 2×2 2 \times 2 integer matrices whose entries aij(1)a_{ij}(1) are all squares of integers and, (2)(2) satisfy aij≤200a_{ij} \le 200. Show that SS has more than 50387(=154−152−15+2) 50387 (=15^4-15^2-15+2) elements, then it has two elements that commute.
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