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For every vector there is a scalar n [ILL 1977]

Source:

January 11, 2011

vectoranalytic geometrygeometry unsolvedgeometry

Problem Statement

A circle

K

centered at

(0,0)

is given. Prove that for every vector

(a_1,a_2)

there is a positive integer

n

such that the circle

K

translated by the vector

n(a_1,a_2)

contains a lattice point (i.e., a point both of whose coordinates are integers).