2014 USAMO 第 3 题

Prove that there exists an infinite set of points

$$\ldots,\,\,\,\,P_{-3},\,\,\,\,P_{-2},\,\,\,\,P_{-1},\,\,\,\,P_0,\,\,\,\,P_1,\,\,\,\,P_2,\,\,\,\,P_3,\,\,\,\,\ldots$$

in the plane with the following property: For any three distinct integers $a,b,$ and $c$, points $P_a$, $P_b$, and $P_c$ are collinear if and only if $a+b+c=2014$.

证明存在无限个点集

$$

\l点,\,\,\,\,P_{-3},\,\,\,\,P_{-2},\,\,\,\,P_{-1},\,\,\,\,P_0,\,\,\,\,P_1,\,\,\,\,P_2,\,\,\,\,P_3,\,\,\,\,\l点

$$

在具有以下属性的平面中：对于任意三个不同的整数 $a,b,$ 和 $c$，点 $P_a$、$P_b$ 和 $P_c$ 共线当且仅当 $a+b+c=2014$。