Problem:
For real number let be the greatest integer less than or equal to , and define to be the fractional part of . For example, and . Define , and let be the number of real-valued solutions to the equation for . Find the remainder when is divided by .
Solution:
Let's look at Suppose for some integer n.
Notice how since increases on the interval , and also increases on the interval
that would also increase on this interval.
Now, let's examine the values that f takes on this interval more closely.
Notice that if we look at on the interval , that
Where x takes values on .
This function is unique as on this interval, can take on any value from to
Thus, notice the way that would have 1 solution, between , one between all the way to
So the number of solutions to can be determined because there's one solution for ways.
The unique part of this is that when examining
, that it can be determined uniquely based off the values of that would be some integer in the range , follows by the value of , which would be another integer that is at most as large as .
Thus, for , we basically can find the number of solutions based off setting 3 values, particularly the value of , under the condition that
This can be solved using stars and bars as the possible floors can be taken on integers based off the list .
Suppose is the number of times 17 is chosen, and define through similarly.
We have to solve the number of solutions to
The problems on this page are the property of the MAA's American Mathematics Competitions