1996 AIME Problem 2

Question

For each real number

x,

let

\lfloor x\rfloor

denote the greatest integer that does not exceed

x.

For how many positive integers

n

is it true that

n<1000

and that

\lfloor\log_2 n\rfloor

is a positive even integer?

Answer

Translate each possible value of

\lfloor\log_2n\rfloor

into a power-of-two interval

The possible even values below

1000

are

2,

4,

6,

and

8

If

\lfloor\log_2n\rfloor=k,

then

2^k\leq n<2^{k+1}.

The value of

k

can be

2,

4,

6,

or

8,

since

2^{10}>1000.

The corresponding interval sizes are

2^2,

2^4,

2^6,

and

2^8.

Therefore the requested number is

4+16+64+256=340.

Problem 2 in other years