2011 AMC 10B Problem 10

Question

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

The denominator is a geometric sum

1+10+\cdots+10^9

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

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 .

Problem 10 in other years