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

Answer: 340

Concepts:floor and ceiling functions logarithm counting integers in a range

Difficulty rating: 1740

Small Hint:

Translate each possible value of $\lfloor\log_2n\rfloor$ into a power-of-two interval

Big Hint:

The possible even values below $1000$ are $2,$ $4,$ $6,$ and $8$

Solution:

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 $4+16+64+256=340.$

1996 AIME Problem 2

Answer: 340

Difficulty rating: 1740

Small Hint:

Big Hint:

Solution:

Problem 2 in Other Years