AMC8 2023

AMC8 2023 · Q16

AMC8 2023 · Q16. It mainly tests Combinatorial geometry (counting), Sequences in number theory (remainders patterns).

The letters $\text{P}, \text{Q},$ and $\text{R}$ are entered into a $20\times20$ 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?

字母 $\text{P}, \text{Q},$ 和 $\text{R}$ 按照下面所示的模式填入一个 $20\times20$ 表格中。完成的表格中会出现多少个 $\text{P}$、$\text{Q}$ 和 $\text{R}$？

(A) 132\text{ Ps, }134\text{ Qs, }134\text{ Rs} 132\text{ Ps, }134\text{ Qs, }134\text{ Rs}

(B) 133\text{ Ps, }133\text{ Qs, }134\text{ Rs} 133\text{ Ps, }133\text{ Qs, }134\text{ Rs}

(D) 134\text{ Ps, }132\text{ Qs, }134\text{ Rs} 134\text{ Ps, }132\text{ Qs, }134\text{ Rs}

(E) 134\text{ Ps, }133\text{ Qs, }133\text{ Rs} 134\text{ Ps, }133\text{ Qs, }133\text{ Rs}

Answer

Correct choice: (C)

正确答案：（C）

Solution

In our $5\times5$ grid, there are $8,9$ and $8$ of the letters $\text{P}, \text{Q},$ and $\text{R}$, respectively, and in a $2\times2$ grid, there are $1,2$ and $1$ of the letters $\text{P}, \text{Q},$ and $\text{R}$, respectively. We see that in both grids, there are $x, x+1,$ and $x$ of the $\text{P}, \text{Q},$ and $\text{R}$, respectively. This is because in any $n\times n$ grid with $n\equiv2\pmod3$, there are $x, x+1,$ and $x$ of the $\text{P}, \text{Q},$ and $\text{R}$, respectively. We can see that the only answer choice which satisfies this condition is $\boxed{\textbf{(C)}~133\text{ Ps, }134\text{ Qs, }133\text{ Rs}}.$

在我们的 $5\times5$ 网格中，字母 $\text{P}, \text{Q},$ 和 $\text{R}$ 分别有 $8,9$ 和 $8$ 个，在 $2\times2$ 网格中，分别有 $1,2$ 和 $1$ 个。我们看到在两个网格中，$\text{P}, \text{Q},$ 和 $\text{R}$ 分别有 $x, x+1,$ 和 $x$ 个。这是因为在任何 $n\times n$ 网格中，当 $n\equiv2\pmod3$ 时，$\text{P}, \text{Q},$ 和 $\text{R}$ 分别有 $x, x+1,$ 和 $x$ 个。我们可以看到唯一满足这个条件的答案选项是 $\boxed{\textbf{(C)}~133\text{ Ps, }134\text{ Qs, }133\text{ Rs}}.$

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