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Letters in rows and columns overload inequality

Source: 2022 Bulgarian Spring Math Competition, Problem 11.3

March 27, 2022
combinatoricstableinequalities

Problem Statement

In every cell of a table with nn rows and mm columns is written one of the letters aa, bb, cc. Every two rows of the table have the same letter in at most k≥0k\geq 0 positions and every two columns coincide at most kk positions. Find mm, nn, kk if 2mn+6k3(m+n)≥k+1\frac{2mn+6k}{3(m+n)}\geq k+1
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