AMC10 2005 A
AMC10 2005 A · Q13
AMC10 2005 A · Q13. It mainly tests Linear inequalities, Manipulating equations.
How many positive integers $n$ satisfy the following condition: $(130n)^{50} > n^{100} > 2^{200}$?
有多少正整数 $n$ 满足条件：$(130n)^{50} > n^{100} > 2^{200}$？
(A)
0
0
(B)
7
7
(C)
12
12
(D)
65
65
(E)
125
125
Answer
Correct choice: (E)
正确答案：（E）
Solution
(E) The condition is equivalent to
$130n>n^2>2^4=16,$ so $130n>n^2$ and $n^2>16.$
This implies that $130>n>4.$ So $n$ can be any of the 125 integers strictly between 130 and 4.
（E）该条件等价于
$130n>n^2>2^4=16,$ 因此 $130n>n^2$ 且 $n^2>16.$
这意味着 $130>n>4.$ 所以 $n$ 可以是严格介于 130 与 4 之间的 125 个整数中的任意一个。
Topics
Related Questions
Practice full AMC exams on amcdrill.
Try full-length practice and diagnostics at www.amcdrill.com.