1985 AIME Problems/Problem 10
Problem
How many of the first 1000 positive integers can be expressed in the form
,
where is a real number, and
denotes the greatest integer less than or equal to
?
Solution
We will be able to reach the same number of integers while ranges from 0 to 1 as we will when
ranges from
to
for any integer
. Since
, the answer must be exactly 50 times the number of integers we will be able to reach as
ranges from 0 to 1, including 1 but excluding 0.
As we change the value of , the value of our expression changes only when
crosses rational number of the form
, where
is divisible by 2, 4, 6 or 8. Thus, we need only see what happens at the numbers of the form
. This gives us 24 calculations to make; we summarize the results here:
Thus, we hit 12 of the first 20 integers and so we hit of the first 100.