Question

设 $T$ 是坐标平面中顶点为 $(0, 0)$ 、 $(4, 0)$ 和 $(0, 3)$ 的三角形。考虑平面上的以下五个等距变换（刚性变换）：绕原点逆时针旋转 $90^\circ$ 、 $180^\circ$ 和 $270^\circ$ ，关于 $x$ 轴反射，以及关于 $y$ 轴反射。在这五种变换中任选三个组成的 $125$ 个序列（不要求互不相同）中，有多少个会把 $T$ 送回原来的位置？（例如，先旋转 $180^\circ$ ，再关于 $x$ 轴反射，再关于 $y$ 轴反射，会把 $T$ 送回原来的位置；但先旋转 $90^\circ$ ，再关于 $x$ 轴反射，再次关于 $x$ 轴反射，则不会把 $T$ 送回原来的位置。）

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：变换分类讨论

难度评级：1910

解答：

因为 $T$ 是不等边直角三角形，唯一将 $T$ 映到自身的等距变换是恒等变换，所以一个序列可行，当且仅当三个变换的复合是恒等变换。

令 $r$ 表示旋转 $90^\circ$ ，令 $s$ 表示关于 $x$ 轴的反射。五个允许的映射是 $r,r^2,r^3,s,r^2s;$ 缺少的非恒等映射是两个对角线反射 $rs,r^3s.$ 在有序三元组中，第三个映射由前两个唯一决定；它被允许，当且仅当前两个的乘积是这五个映射之一。

恰有 $5$ 个有序对的乘积是恒等变换：每个第一个映射都与它的逆映射配对。得到对角线反射需要一个旋转和一个轴反射； $r$ 或 $r^3$ 可以在左侧或右侧与 $s$ 或 $r^2s,$ 配对，共有 $2\cdot2\cdot2=8$ 对。剩余 $25-5-8=12$ 个有序对给出有效序列。

所以 A 是正确答案。

Because $T$ is a scalene right triangle, the only isometry carrying $T$ to itself is the identity, so a sequence works exactly when the three transformations compose to the identity.

Let $r$ be the $90^\circ$ rotation and $s$ reflection across the $x$ -axis. The five allowed maps are $r,r^2,r^3,s,r^2s;$ the missing nonidentity maps are the diagonal reflections $rs,r^3s.$ In an ordered triple, the third map is forced by the first two and is allowed precisely when their product is one of the five.

Exactly $5$ ordered pairs have product identity: each first map is paired with its inverse. A diagonal reflection requires one rotation and one axis reflection; $r$ or $r^3$ may be paired on either side with $s$ or $r^2s,$ giving $2\cdot2\cdot2=8$ pairs. The remaining $25-5-8=12$ ordered pairs give valid sequences.

Thus, A is the correct answer.

2020 AMC 12A 第 20 题

答案：A

难度评级：1910

解答：

其他年份的第 20 题