Question

字母 P、Q、R 按下图所示的规律填入一个 $20 \times 20$ 表格。完成后的表格中会出现多少个 P、Q 和 R？

The letters P, Q, and R are entered into a $20 \times 20$ table according to the pattern shown below. How many Ps, Qs, and Rs will appear in the completed table?

$132$ 个 P， $134$ 个 Q， $134$ 个 R

$132$ Ps, $134$ Qs, $134$ Rs

$133$ 个 P， $133$ 个 Q， $134$ 个 R

$133$ Ps, $133$ Qs, $134$ Rs

$133$ 个 P， $134$ 个 Q， $133$ 个 R

$133$ Ps, $134$ Qs, $133$ Rs

$134$ 个 P， $132$ 个 Q， $134$ 个 R

$134$ Ps, $132$ Qs, $134$ Rs

$134$ 个 P， $133$ 个 Q， $133$ 个 R

$134$ Ps, $133$ Qs, $133$ Rs

答案：C

知识点：找规律模运算

难度评级：1540

视频讲解：

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文字解答：

因为 $20 = 3 \cdot 6 + 2$ ，每列底部的 $2$ 个字母会比第三个字母多出现一次。

所以每列中第三个字母出现 $6$ 次，另外 $2$ 个字母各出现 $7$ 次。

同样分析可知， $R$ 和 $P$ 在 $7$ 列中处于第三个位置，而 $Q$ 只在这些列中出现 $6$ 次。

因此在 $20 - 7 = 13$ 列中， $P$ 和 $R$ 各出现 $7$ 次，共 $13 \cdot 7 = 91$ 次。

它们还会在另外 $7$ 列中各出现 $6$ 次，增加 $6 \cdot 7 = 42$ 次。因此 $P$ 和 $R$ 各出现 $91 + 42 = 133$ 次。

$Q$ 的出现次数为 $\begin{align*} 20 \cdot 20 - 2 \cdot 133 &=400 - 266 \\ &= 134 \end{align*}$

所以正确答案是 C。

Since $20 = 3 \cdot 6 + 2,$ the bottom $2$ letters in each column will occur one more time than the third letter.

This means that the third letter in each column will occur $6$ times, whereas the other $2$ will appear $7$ times.

Using the same analysis, we can see that $R$ and $P$ appear in the third row $7$ times, whereas $Q$ only appears $6$ times.

Therefore, in $20 - 7 = 13$ columns, $P$ and $R$ appear $7$ times, for a total of $13 \cdot 7 = 91$ times.

They also appear $6$ times in $7$ columns, adding $6 \cdot 7 = 42$ appearances to the total. This gives us a total of $91 + 42 = 133$ appearances for $P$ and $R.$

$Q$ will therefore appear $\begin{align*} 20 \cdot 20 - 2 \cdot 133 &=400 - 266 \\ &= 134 \end{align*}$ times.

Thus, C is the correct answer.

2023 AMC 8 第 16 题

答案：C

难度评级：1540

视频讲解：

文字解答：

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