Question

若平面上的点 $(x,y)$ 的两个坐标 $x$ 和 $y$ 都是整数，则称它为格点。内部恰好含三个格点的最大正方形，其面积最接近

A point $(x,y)$ in the plane is called a lattice point if both $x$ and $y$ are integers. The area of the largest square that contains exactly three lattice points in its interior is closest to

$4.0$

$4.2$

$4.5$

$5.0$

$5.6$

答案：D

知识点：lattice points extremal geometry

难度评级：2520

小提示：

三个共线的内部格点会迫使出现第四个，所以应使用面积最小的非共线格点三角形

Three collinear interior lattice points force a fourth, so use a smallest noncollinear lattice triangle

大提示：

正方形达到最大时，相对的两条边被附近格点卡住；比较这些平行线之间的距离

At a maximal square, opposite sides are pinned by nearby lattice points; compare their separation

解答：

只需包围三个非共线格点 $(0,0),(0,1),(1,0)$ 。在最大位置，相对的两条边被邻近格点卡住；最大可能距离为 $\sqrt5$ ，对应的两条平行线分别经过 $(1,1)$ 与 $(0,-1)$ 。因此面积至多为 $5$ 。由下列四条直线围成的正方形达到这个上界： $\begin{aligned} y&=2x+2,\\ y&=2x-3,\\ 2y+x&=3,\\ 2y+x&=-2 \end{aligned}\text{。}$ 其面积为 $5$ ，且内部恰好包含上述三个格点。所以正确答案是 D。

It suffices to enclose the three noncollinear lattice points $(0,0),(0,1),(1,0).$ In a maximal placement, two opposite sides are pinned by neighboring lattice points; the greatest possible separation is the distance $\sqrt5$ between parallel lines through $(1,1)$ and $(0,-1).$ Hence the area is at most $5.$ This is attained by the square bounded by $\begin{aligned} y&=2x+2,\\ y&=2x-3,\\ 2y+x&=3,\\ 2y+x&=-2. \end{aligned}$ Its area is $5$ and exactly the three stated lattice points lie inside. Thus D is correct.

1998 AMC 12 第 29 题

答案：D

难度评级：2520

小提示：

大提示：

解答：

其他年份的第 29 题