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Variable point M - ILL 1990 PRK2

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September 18, 2010
analytic geometrygeometry

Problem Statement

In coordinate plane, a variable point MM, starting from the origin O(0,0)O(0, 0), moves on the line ll with slope kk, where kk is an irrational number.
(i) Prove that point O(0,0)O(0, 0) is the only rational point (namely, the coordinates of which are both rationals) on the line l.l.
(ii) Prove that for any number ε>0\varepsilon > 0, there exist integers m,nm, n such that the distance between ll and the point (m,n)(m, n) is less than ε.\varepsilon.
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