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max AP/PE in lattice ZxZ, unique P lies interior to ABC

Source: Stars O f Mathematics 2009 p3 - 1998 Moldova TST p11

May 28, 2019
lattice pointslatticemaximumratiogeometry

Problem Statement

Let A,B,CA,B,C be nodes of the lattice Z×ZZ\times Z such that inside the triangle ABCABC lies a unique node PP of the lattice. Denote E=AP∩BCE = AP \cap BC. Determine max APPE\frac{AP}{PE} , over all such configurations.
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