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Points Collinear iff Sum is Constant

Source: USAMO 2014, Problem 3

April 29, 2014
analytic geometrylinear algebramatrixgraphing linespolynomial

Problem Statement

Prove that there exists an infinite set of points …,  P−3,  P−2,  P−1,  P0,  P1,  P2,  P3,  … \dots, \; P_{-3}, \; P_{-2},\; P_{-1},\; P_0,\; P_1,\; P_2,\; P_3,\; \dots in the plane with the following property: For any three distinct integers a,b,a,b, and cc, points PaP_a, PbP_b, and PcP_c are collinear if and only if a+b+c=2014a+b+c=2014.
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