1998 AMC 12 第 29 题

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

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

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

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.

其他年份的第 29 题