Question

有多少个边平行于坐标轴、顶点坐标都是整数的正方形完全位于由直线 $y=\pi x$ 、直线 $y=-0.1$ 和直线 $x=5.1$ 围成的区域内？

How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x,$ the line $y=-0.1$ and the line $x=5.1?$

$\ 30$

$\ 41$

$\ 45$

$\ 50$

$\ 57$

答案：D

知识点：格点图形中的形状计数分类讨论

难度评级：1970

解答：

正方形必须在 $y=-0.1$ 上方、 $x=5.1$ 左方、且在 $y=\pi x$ 下方。因为 $\pi$ 略大于 $3$ ，在 $x=1,2,3,4$ 处可用的整数高度分别是 $3,6,9,12$ ，边长大于 $3$ 的正方形不能完全放入。

按左上顶点和边长计数。边长为 $1$ 时有 $3+6+9+12=30$ 个；边长为 $2$ 时有 $2+5+8=15$ 个；边长为 $3$ 时有 $1+4=5$ 个。

总数为 $30+15+5=50$ 。

所以正确答案是 D。

A square must lie above $y=-0.1$ , to the left of $x=5.1$ , and below $y=\pi x$ . Since $\pi$ is a little more than $3$ , the lattice heights available at $x=1,2,3,4$ are $3,6,9,12$ , and side lengths larger than $3$ cannot fit.

Count by side length using the top-left lattice point. For side length $1$ , there are $3+6+9+12=30$ choices. For side length $2$ , there are $2+5+8=15$ choices. For side length $3$ , there are $1+4=5$ choices.

The total is $30+15+5=50$ .

Thus, the correct answer is D.

2016 AMC 10B 第 14 题

答案：D

难度评级：1970

解答：

其他年份的第 14 题