2014 AMC 12A Problem 22

Question

The number

5^{867}

is between

2^{2013}

and

2^{2014}.

How many pairs of integers

(m,n)

are there such that

1\le m\le2012

and

5^n\lt2^m\lt2^{m+2}\lt5^{n+1}?

278

279

280

281

282

Between consecutive powers

5^n

and

5^{n+1}

there are either two or three powers of

2

The inequality asks for three such powers; if

d,t

count the two-power and three-power gaps, then

d+t=867

and

2d+3t=2013

Because

2^2\lt5\lt2^3,

each interval

(5^n,5^{n+1})

contains either two or three powers of

2.

The chain

5^n\lt2^m\lt2^{m+2}\lt5^{n+1}

holds exactly when the interval contains three consecutive powers of

2,

and then there is a unique such

m.

Let

d

and

t

be the numbers of intervals

(5^n,5^{n+1})

for

0\le n\le866

containing two and three powers of

2,

respectively. Since

2^{2013}\lt5^{867}\lt2^{2014}

there are

2013

powers of

2

in total, giving

d+t=867

and

2d+3t=2013.

Solving,

t=2013-2\cdot867=279.

Thus, the correct answer is B.

Problem 22 in other years