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min distance of lattice point from 5x-10y+3=0 is \sqrt3/ 20

Source: 1985 Greece MO Grade XII p3

September 6, 2024
analytic geometrygeometrydistancenumber theorylattice points

Problem Statement

Consider the line (E): 5x−10y+3=05x-10y+3=0 . Prove that: a) Line (E)(E) doesn't pass through points with integer coordinates. b) There is no point A(a1,a2)A(a_1,a_2) with a1,a2∈Z a_1,a_2 \in \mathbb{Z} with distance from (E)(E) less then 320\frac{\sqrt3}{20}.
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