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maximum area of lattice triangle containing exactly one m-lattice

Source: 2021ChinaTST test3 day2 P3

April 13, 2021
geometrycombinatoricscombinatorial geometry

Problem Statement

Proof that there exist constant λ\lambda, so that for any positive integer m(≥2)m(\ge 2), and any lattice triangle TT in the Cartesian coordinate plane, if TT contains exactly one mm-lattice point in its interior(not containing boundary), then TT has area ≤λm3\le \lambda m^3. PS. lattice triangles are triangles whose vertex are lattice points; mm-lattice points are lattice points whose both coordinates are divisible by mm.
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