AMC10 2014 A

AMC10 2014 A · Q25

AMC10 2014 A · Q25. It mainly tests Systems of equations, Powers & residues.

The number 5867 is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m, n)$ are there such that $1 \le m \le 2012$ and $5^n < 2^m < 2^{m+2} < 5^{n+1}$?

数 5867 位于 $2^{2013}$ 和 $2^{2014}$ 之间。有多少对整数 $(m, n)$ 满足 $1 \le m \le 2012$ 且 $5^n < 2^m < 2^{m+2} < 5^{n+1}$？

(A) 278 278

(B) 279 279

(D) 281 281

(E) 282 282

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): Because $2^2<5$ and $2^3>5$, there are either two or three integer powers of 2 strictly between any two consecutive integer powers of 5. Thus for each $n$ there is at most one $m$ satisfying the given inequalities, and the question asks for the number of cases in which there are three powers rather than two. Let $d$ (respectively, $t$) be the number of nonnegative integers $n$ less than 867 such that there are exactly two (respectively, three) powers of 2 strictly between $5^n$ and $5^{n+1}$. Because $2^{2013}<5^{867}<2^{2014}$, it follows that $d+t=867$ and $2d+3t=2013$. Solving the system yields $t=279$.

答案（B）：因为$2^2<5$且$2^3>5$，在任意两个相邻的 5 的整数次幂之间，严格夹着的 2 的整数次幂要么有 2 个，要么有 3 个。因此对每个$n$，至多有一个$m$满足所给不等式，而题目要求的是出现 3 个幂而不是 2 个幂的情形数。设$d$（分别地，$t$）为小于 867 的非负整数$n$的个数，使得在$5^n$与$5^{n+1}$之间严格夹着的 2 的幂恰有 2 个（分别地，3 个）。因为$2^{2013}<5^{867}<2^{2014}$，可得$d+t=867$且$2d+3t=2013$。解该方程组得$t=279$。

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