Question

设 $T$ 为坐标平面中顶点为 $(0,0), (4,0)$ 和 $(0,3)$ 的三角形。考虑平面上的下列五个等距变换：绕原点逆时针旋转 $90^{\circ}, 180^{\circ}$ 和 $270^{\circ}$ ，关于 $x$ 轴反射，关于 $y$ 轴反射。在这五个变换中选取三个组成序列，允许重复，共 $125$ 个序列，其中有多少个会把 $T$ 变回原来的位置？例如，先旋转 $180^{\circ}$ ，再关于 $x$ 轴反射，再关于 $y$ 轴反射，会把 $T$ 变回原位；但先旋转 $90^{\circ}$ ，再关于 $x$ 轴反射，再关于 $x$ 轴反射，则不会把 $T$ 变回原位。

Let $T$ be the triangle in the coordinate plane with vertices $(0,0), (4,0),$ and $(0,3).$ Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}, 180^{\circ},$ and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$ -axis, and reflection across the $y$ -axis. How many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position? (For example, a $180^{\circ}$ rotation, followed by a reflection across the $x$ -axis, followed by a reflection across the $y$ -axis will return $T$ to its original position, but a $90^{\circ}$ rotation, followed by a reflection across the $x$ -axis, followed by another reflection across the $x$ -axis will not return $T$ to its original position.)

$12$

$15$

$17$

$20$

$25$

答案：A

知识点：变换系统列举

难度评级：1950

视频讲解：

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文字解答：

令 $R$ 表示 $90^\circ$ 旋转，允许的旋转为 $R,R^2,R^3$ 。令 $X$ 和 $Y$ 分别表示关于两个坐标轴的反射。前两个变换确定后，第三个变换必须是它们乘积的逆，才能使总效果为恒等变换。

枚举前两个变换，可行的有序对为 $6$ $R^2$ $4$ $2$ $6+4+2=12$ ，以及。共有个序列。正确答案是 A。

Let $R$ be a $90^\circ$ rotation, so the allowed rotations are $R,R^2,R^3$ . Let $X$ and $Y$ be the reflections across the coordinate axes. Once the first two transformations are chosen, the third is forced to be the inverse of their product.

Among two rotations, $6$ ordered pairs have a nonidentity rotation as their product. A rotation and a reflection have an allowed axis-reflection as their product exactly when the rotation is $R^2$ , giving $4$ ordered pairs. Finally, the two different axis-reflections can occur in either order, giving $2$ more pairs. Altogether there are $6+4+2=12$ valid sequences. Thus, A is the correct answer.

2020 AMC 10A 第 23 题

答案：A

难度评级：1950

视频讲解：

文字解答：

其他年份的第 23 题