AMC12 2014 A

AMC12 2014 A · Q22

AMC12 2014 A · Q22. It mainly tests Counting & probability misc, Remainders & modular arithmetic.

The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and \[5^n<2^m<2^{m+2}<5^{n+1}?\]

数 $5^{867}$ 位于 $2^{2013}$ 和 $2^{2014}$ 之间。有多少对整数 $(m,n)$ 满足 $1\leq m\leq 2012$ 且 \[5^n<2^m<2^{m+2}<5^{n+1}?\]

(A) 278 278

(B) 279 279

(D) 281 281

(E) 282\qquad 282\qquad

Answer

Correct choice: (B)

正确答案：（B）

Solution

Between any two consecutive powers of $5$ there are either $2$ or $3$ powers of $2$ (because $2^2<5^1<2^3$). Consider the intervals $(5^0,5^1),(5^1,5^2),\dots (5^{866},5^{867})$. We want the number of intervals with $3$ powers of $2$. From the given that $2^{2013}<5^{867}<2^{2014}$, we know that these $867$ intervals together have $2013$ powers of $2$. Let $x$ of them have $2$ powers of $2$ and $y$ of them have $3$ powers of $2$. Thus we have the system \[x+y=867\]\[2x+3y=2013\] from which we get $y=279$, so the answer is $\boxed{\textbf{(B)}}$.

在任意两个连续的 $5$ 的幂之间，有 $2$ 或 $3$ 个 $2$ 的幂（因为 $2^2<5^1<2^3$）。考虑区间 $(5^0,5^1),(5^1,5^2),\dots (5^{866},5^{867})$。我们要求有 $3$ 个 $2$ 的幂的区间个数。由给定 $2^{2013}<5^{867}<2^{2014}$，知这 $867$ 个区间总共有 $2013$ 个 $2$ 的幂。设其中 $x$ 个区间有 $2$ 个 $2$ 的幂，$y$ 个有 $3$ 个，则有方程组 \[x+y=867\]\[2x+3y=2013\] 解得 $y=279$，故答案为 $\boxed{\textbf{(B)}}$。

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