AMC12 2016 B

AMC12 2016 B · Q11

AMC12 2016 B · Q11. It mainly tests Coordinate geometry, Counting in geometry (lattice points).

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 $x=5.1$?

有多少个边与坐标轴平行、且顶点坐标都是整数的正方形，能够完全位于由直线 $y=\pi x$、直线 $y=-0.1$ 以及直线 $x=5.1$ 所围成的区域内？

(A) 30 30

(B) 41 41

(D) 50 50

(E) 57 57

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): Note that $3<\pi<4$, $6<2\pi<7$, $9<3\pi<10$, and $12<4\pi<13$. Therefore there are $3$ 1-by-1 squares of the desired type in the strip $1\le x\le 2$, $6$ 1-by-1 squares in the strip $2\le x\le 3$, $9$ 1-by-1 squares in the strip $3\le x\le 4$, and $12$ 1-by-1 squares in the strip $4\le x\le 5$. Furthermore there are $2$ 2-by-2 squares in the strip $1\le x\le 3$, $5$ 2-by-2 squares in the strip $2\le x\le 4$, and $8$ 2-by-2 squares in the strip $3\le x\le 5$. There is $1$ 3-by-3 square in the strip $1\le x\le 4$, and there are $4$ 3-by-3 squares in the strip $2\le x\le 5$. There are no 4-by-4 or larger squares. Thus in all there are $3+6+9+12+2+5+8+1+4=50$ squares of the desired type within the given region.

答案（D）：注意 $3<\pi<4$，$6<2\pi<7$，$9<3\pi<10$，以及 $12<4\pi<13$。因此，在条带 $1\le x\le 2$ 中有 $3$ 个所求类型的 $1\times1$ 正方形，在条带 $2\le x\le 3$ 中有 $6$ 个 $1\times1$ 正方形，在条带 $3\le x\le 4$ 中有 $9$ 个 $1\times1$ 正方形，在条带 $4\le x\le 5$ 中有 $12$ 个 $1\times1$ 正方形。此外，在条带 $1\le x\le 3$ 中有 $2$ 个 $2\times2$ 正方形，在条带 $2\le x\le 4$ 中有 $5$ 个 $2\times2$ 正方形，在条带 $3\le x\le 5$ 中有 $8$ 个 $2\times2$ 正方形。在条带 $1\le x\le 4$ 中有 $1$ 个 $3\times3$ 正方形，而在条带 $2\le x\le 5$ 中有 $4$ 个 $3\times3$ 正方形。不存在 $4\times4$ 或更大的正方形。因此，总共有 $3+6+9+12+2+5+8+1+4=50$ 个所求类型的正方形位于给定区域内。

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