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Problems(1)

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12/23/2023
Let RR be the rectangle on the cartesian plane with vertices (0,0)(0,0), (5,0)(5,0), (5,7)(5,7), and (0,7)(0,7). Find the number of squares with sides parallel to the axes and vertices that are lattice points that lie within the region bounded by RR.
Proposed by Boyan Litchev
Solution. 85\boxed{85} We have (6−n)(8−n)(6-n)(8-n) distinct squares with side length nn, so the total number of squares is 5⋅7+4⋅6+3⋅5+2⋅4+1⋅3=855 \cdot 7+4 \cdot 6+3 \cdot 5+2 \cdot 4+1\cdot 3 = \boxed{85}.
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