AMC 8 2023 Problem 16

AMC8 2023 年第 16 题

数列与规律★★★☆☆

The letters

\text{P}

\text{Q}

, and

\text{R}

are entered into a

20\times 20

table according to the pattern shown below. How many

\text{P}

's,

\text{Q}

's, and

\text{R}

's will appear in the completed table?

将字母

\text{P}

、

\text{Q}

、

\text{R}

按下图所示的规律填入

20\times 20

的方格表中。问在完成的表中，各会出现多少个

\text{P}

、多少个

\text{Q}

、多少个

\text{R}

？

正确答案 · Correct Answer

解析 · Solution

Solution 1

Since

20\equiv2\pmod3

and

\text{Q}

is in the 2nd diagonal, it is also in the 20th diagonal, and so we find that there are

2(2 + 5 + 8 + 11 + 14 + 17) + 20 = 134 \text{Qs}

. Since all the

\text{P}

's and

\text{R}

's are symmetric, the answer is

text{ Ps,134\text{ Qs, }133\text{ Rs}}.

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