Question

考虑集合 $\{1, 10, 10^2, 10^3, \ldots, 10^{10}\}$ 。该集合最大元素与其他十个元素之和的比值最接近哪个整数？

Consider the set of numbers $\{1, 10, 10^2, 10^3, \ldots, 10^{10}\}.$ The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer?

$1$

$9$

$10$

$11$

$101$

答案：B

知识点：等比数列估算

难度评级：1280

解答：

最大元素为 $10^{10}$ 。其余元素之和为于是 $9$ ，从而。因此接近一，与其余元素之和的比值接近 $9$ 。 $S=\sum_{i=0}^9 10^i=\dfrac{10^{10}-1}{9}.$ $\dfrac{10^{10}}{S}=9\cdot\dfrac{10^{10}}{10^{10}-1},$

所以正确答案是 B。

The largest number is $10^{10}.$ The other ten numbers have sum $S=\sum_{i=0}^9 10^i=\dfrac{10^{10}-1}{9}.$ Therefore their ratio is $\dfrac{10^{10}}{S}=9\cdot\dfrac{10^{10}}{10^{10}-1},$ which is just slightly greater than $9$ and hence closest to $9.$

Thus, the correct answer is B .

2011 AMC 10B 第 10 题

答案：B

难度评级：1280

解答：

其他年份的第 10 题