Question

若一个正实数存在只由数字 $0$ 和 $7$ 组成的十进制表示，就称它为特殊数。例如， $\frac{700}{99}=7.07=7.070707\ldots$ 和 $77.007$ 都是特殊数。求最小的 $n$ ，使 $1$ 可以表示为 $n$ 个特殊数之和。

Call a positive real number special if it has a decimal representation that consists entirely of digits $0$ and $7.$ For example, $\frac{700}{99}=7.07=7.070707\ldots$ and $77.007$ are special numbers. What is the smallest $n$ such that $1$ can be written as a sum of $n$ special numbers?

$7$

$8$

$9$

$10$

$1$ 不能表示为有限个特殊数之和

$1$ cannot be represented as a sum of finitely many special numbers

答案：B

知识点：decimal expansions 位值 construction

难度评级：2410

小提示：

若有 $a_k$ 个加数在第 $k$ 个小数位上是 $7$ ，将等式两边除以 $7$

If $a_k$ summands have $7$ in decimal place $k$ , divide the sum by $7$

大提示：

将所得的各位计数与 $\frac{1}{7}$ 的循环小数比较，再寻找一个以六位为周期的构造

Compare the resulting digit counts with the repeating decimal for $\frac{1}{7}$ , then seek a six-digit repeating construction

解答：

假设 $1$ 是 $n$ 个特殊数之和，并令 $a_k$ 表示在第 $k$ 个小数位上取数字 $7$ 的加数个数。将等式两边除以 $7$ ，可得 $\begin{aligned} \frac17&=\frac{a_1}{10}+\frac{a_2}{10^2}+\cdots\\ &=0.\overline{142857} \end{aligned}\text{。}$ 当 $n\le9$ 时，每个 $a_k$ 都是一位数字，所以 $a_1,a_2,\ldots=1,4,2,8,5,7,\ldots$ ；因此 $n\ge8$ 。八个就足够，因为以下六位循环块所表示的特殊循环小数满足 $\begin{aligned} 700700+2(070707)\\ {}+2(077777)\\ {}+3(000777)&=999999 \end{aligned}\text{。}$ 它们的和为 $1$ 。因此最小值为 $8$ ，正确答案是 B。

Suppose $1$ is a sum of $n$ special numbers, and let $a_k$ count summands having a $7$ in the $k$ th decimal place. Dividing by $7$ gives $\begin{aligned} \frac17&=\frac{a_1}{10}+\frac{a_2}{10^2}+\cdots\\ &=0.\overline{142857}. \end{aligned}$ For $n\le9,$ each $a_k$ is a digit, so $a_1,a_2,\ldots=1,4,2,8,5,7,\ldots;$ hence $n\ge8.$ Eight suffice because the repeating special decimals represented by $\begin{aligned} 700700+2(070707)\\ {}+2(077777)\\ {}+3(000777)&=999999. \end{aligned}$ Their sum is $1.$ Therefore the minimum is $8,$ and B is correct.

1997 AMC 12 第 29 题

答案：B

难度评级：2410

小提示：

大提示：

解答：

其他年份的第 29 题