AMC10 2016 B

AMC10 2016 B · Q14

AMC10 2016 B · Q14. It mainly tests Basic counting (rules of product/sum), Coordinate geometry.

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、且顶点坐标均为整数的正方形，能够完全位于由直线 $y=\pi x$、直线 $y=-0.1$ 和直线 $x=5.1$ 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 $1\le x\le2$, 6 1-by-1 squares in the strip $2\le x\le3$, 9 1-by-1 squares in the strip $3\le x\le4$, and 12 1-by-1 squares in the strip $4\le x\le5$. Furthermore there are 2 2-by-2 squares in the strip $1\le x\le3$, 5 2-by-2 squares in the strip $2\le x\le4$, and 8 2-by-2 squares in the strip $3\le x\le5$. There is 1 3-by-3 square in the strip $1\le x\le4$, and there are 4 3-by-3 squares in the strip $2\le x\le5$. 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，在带状区域 $1\le x\le2$ 中有 3 个所需类型的 $1\times1$ 正方形，在 $2\le x\le3$ 中有 6 个，在 $3\le x\le4$ 中有 9 个，在 $4\le x\le5$ 中有 12 个。此外，在带状区域 $1\le x\le3$ 中有 2 个 $2\times2$ 正方形，在 $2\le x\le4$ 中有 5 个，在 $3\le x\le5$ 中有 8 个。在带状区域 $1\le x\le4$ 中有 1 个 $3\times3$ 正方形，在 $2\le x\le5$ 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 $3+6+9+12+2+5+8+1+4=50$ 个所需类型的正方形位于给定区域内。

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