2011 AMC 10B 第 10 题

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

分母是一个等比数列和。

The denominator is a geometric sum

1+10+\cdots+10^9

=\frac{10^{10}-1}{9}

。

1+10+\cdots+10^9

=\frac{10^{10}-1}{9}

最大元素为

10^{10}

。其余十个元素之和为

S=\sum_{i=0}^9 10^i=\dfrac{10^{10}-1}{9}\text{。}

因此所求比值为

\dfrac{10^{10}}{S}=9\cdot\dfrac{10^{10}}{10^{10}-1}\text{，}

它略大于

9

，所以最接近

9

。

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 .

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