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2020 AMC 12B 第 19 题

先试着解答 2020 AMC 12B 第 19 题，然后核对你的答案与精心整理的解答，解答来自 LIVE by Po-Shen Loh。你也可以参加完整限时模拟考试、查看全部 2020 AMC 12B 解答，或核对答案。

所有题目均经美国数学协会（MAA）官方合法授权使用。

19.

坐标平面中的正方形 ABCDABCD 的顶点为 A(1,1)A(1, 1)、B(−1,1)B(-1, 1)、C(−1,−1)C(-1, -1) 和 D(1,−1)D(1, -1)。考虑以下四个变换：

LL，绕原点逆时针旋转 90∘90^\circ；

RR，绕原点顺时针旋转 90∘90^\circ；

HH，关于 xx-轴反射；

VV，关于 yy-轴反射。

每个变换都会把正方形映到自身，但标记顶点的位置会改变。例如，先做 RR 再做 VV，会把顶点 AA 从 (1,1)(1, 1) 送到 (−1,−1)(-1, -1)，并把顶点 BB 从 (−1,1)(-1, 1) 送回原位。从 {L,R,H,V}\{L, R, H, V\} 中选出长度为 2020 的变换序列，有多少个会把所有标记顶点都送回原位？（例如，RR，RR，VV，HH 是一个长度为 44 的有效序列。）

Square ABCDABCD in the coordinate plane has vertices at the points A(1,1),A(1, 1), B(−1,1),B(-1, 1), C(−1,−1),C(-1, -1), and D(1,−1).D(1, -1). Consider the following four transformations:

L,L, a rotation of 90∘90^\circ counterclockwise around the origin;

R,R, a rotation of 90∘90^\circ clockwise around the origin;

H,H, a reflection across the xx-axis; and

V,V, a reflection across the yy-axis.

Each of these transformations maps the square onto itself, but the positions of the labeled vertices will change. For example, applying RR and then VV would send the vertex AA at (1,1)(1, 1) to (−1,−1)(-1, -1) and would send the vertex BB at (−1,1)(-1, 1) to itself. How many sequences of 2020 transformations chosen from {L,R,H,V}\{L, R, H, V\} will send all of the labeled vertices back to their original positions? (For example, R,R, R,R, V,V, HH is one sequence of 44 transformations that will send the vertices back to their original positions.)

2372^{37}

3⋅2363 \cdot 2^{36}

2382^{38}

3⋅2373 \cdot 2^{37}

2392^{39}

答案：C
知识点：变换分类讨论
难度评级：2000
小提示：

前 1919 个变换可以任意选择；最后一个变换会被唯一地强制为前面合成的逆。

The first 1919 transformations can be anything; the last one is then forced to be the unique inverse

大提示：

这个被强制的最后一步必须是四个允许变换之一；求它属于允许集合的计数。

The forced last move must be one of the four allowed transformations; find the probability that it is

解答：

将顶点依次标为 0,1,2,30,1,2,3。每个允许的变换都形如 j↦εj+δ(mod4)j\mapsto \varepsilon j+\delta\pmod 4，其中 ε∈{1,−1}\varepsilon\in\{1,-1\} 且 δ∈{1,−1}\delta\in\{1,-1\}。这四种选择恰好给出两个四分之一周旋转和题中所述的两个反射。

在复合时，δ\delta 的奇偶性在每一步都会改变。因此，1919 个允许变换的复合仍有奇数的 δ\delta，所以它仍是四个允许变换之一。它的逆变换也被允许。因此，前 1919 步的每个序列都恰有一种最后一步的选择，共得到 419=2384^{19}=2^{38} 个成功序列。

因此，正确答案是 C。

Label the vertices 0,1,2,30,1,2,3 cyclically. Each allowed transformation has the form j↦εj+δ(mod4),j\mapsto \varepsilon j+\delta\pmod 4, where ε∈{1,−1}\varepsilon\in\{1,-1\} and δ∈{1,−1}.\delta\in\{1,-1\}. These four choices give exactly the two quarter-turns and the two stated reflections.

Under composition, the parity of δ\delta changes at every move. Thus a composition of 1919 allowed transformations again has odd δ,\delta, and so is one of the four allowed transformations. Its inverse is also allowed. Consequently every sequence of the first 1919 moves has exactly one choice for the final move, giving 419=2384^{19}=2^{38} successful sequences.

Thus, the correct answer is C.

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