AMC10 2020 A

AMC10 2020 A · Q23

AMC10 2020 A · Q23. It mainly tests Basic counting (rules of product/sum), Transformations.

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.)

让 $T$ 为坐标平面上的三角形，顶点为 $(0, 0)$、$(4, 0)$ 和 $(0, 3)$。考虑平面的以下五种等距变换（刚性变换）：绕原点逆时针旋转 $90^\circ$、$180^\circ$ 和 $270^\circ$，反射过 $x$ 轴，反射过 $y$ 轴。其中 125 种由这些变换组成的三个变换序列（不一定不同）有多少种会使 $T$ 回到原位置？（例如，$180^\circ$ 旋转后反射过 $x$ 轴再反射过 $y$ 轴会使 $T$ 回到原位置，但 $90^\circ$ 旋转后反射过 $x$ 轴再反射过 $x$ 轴不会。）

(A) 12 12

(B) 15 15

(D) 20 20

(E) 25 25

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): If two rotations that are not inverses of each other are applied in order, then there is a unique third rotation that will return $T$ to its original position. There are $3\cdot2=6$ of these. If two different reflections across the coordinate axes are applied in order, then the $180^\circ$ rotation is the unique transformation that will return $T$ to its original position. There are $2$ of these. If the first two transformations are a rotation of $180^\circ$ and a reflection, in either order, then the other reflection is the unique transformation that will return $T$ to its original position. There are $4$ such sequences. It remains to show that for all other choices of the first two transformations, none of the five choices for the third transformation will return $T$ to its original position. Two transformations that are inverses of each other already return $T$ to its original position, so applying a third transformation will move $T$ out of its original position. The other cases are a rotation of $90^\circ$ or $270^\circ$ followed by a horizontal or vertical reflection, or vice versa. In each case, a reflection would be needed to restore the correct orientation of the triangle, but neither a vertical reflection nor a horizontal reflection will return $T$ to its original position. In all, there are $6+2+4=12$ sequences of three of the given transformations that will return $T$ to its original position.

答案（A）：如果依次施加两个互不为逆变换的旋转，那么存在唯一的第三个旋转能使 $T$ 回到原来的位置。这样的情况有 $3\cdot2=6$ 种。如果依次施加关于坐标轴的两种不同反射，那么 $180^\circ$ 旋转是唯一能使 $T$ 回到原来位置的变换。这样的情况有 $2$ 种。如果前两个变换是 $180^\circ$ 旋转与一次反射（顺序任意），那么另一次反射是唯一能使 $T$ 回到原来位置的变换。这样的序列有 $4$ 种。还需说明：对于其余所有前两个变换的选择，在第三步的五种选择中，没有一种能使 $T$ 回到原来的位置。互为逆变换的两个变换本身就已经使 $T$ 回到原来的位置，因此再施加第三个变换会使 $T$ 离开原来的位置。其余情况是：先进行 $90^\circ$ 或 $270^\circ$ 的旋转，再进行水平或垂直反射，或顺序相反。在每一种情况下，都需要一次反射来恢复三角形的正确朝向，但无论是垂直反射还是水平反射都无法使 $T$ 回到原来的位置。总之，在给定的这些变换中，恰有 $6+2+4=12$ 种由三个变换组成的序列能使 $T$ 回到原来的位置。

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