2020 AIME II Problems/Problem 14
Contents
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 1
A combinatorical argument exists which is the official solution, but I will present another here for the sake of variety.
Applying algebraic manipulation and the hockey-stick identity times gives
Solution 2
To solve , we need to solve
where
, and to solve that we need to solve
where
.
It is clear to see for some integer there is exactly one value of
in the interval
where
To understand this, imagine the graph of
on the interval
The graph starts at
, is continuous and increasing, and approaches
. So as long as
, there will be a solution for
in the interval.
Using this logic, we can find the number of solutions to . For every interval
where
there will be one solution for x in that interval. However, the question states
, but because
doesn't work we can change it to
. Therefore,
, and there are
solutions to
.
We can solve similarly.
to satisfy the bounds of
, so there are
solutions to
, and
to satisfy the bounds of
.
Going back to , there is a single solution for z in the interval
, where
. (We now have an upper bound for
because we know
.) There are
solutions for
, and the floors of these solutions create the sequence
Lets first look at the solution of where
. Then
would have
solutions, and the floors of these solutions would also create the sequence
.
If we used the solution of where
, there would be
solutions for
. If we used the solution of
where
, there would be
solutions for
, and so on. So for the solution of
where
, there will be
solutions for
If we now look at the solution of where
, there would be
solutions for
. If we looked at the solution of
where
, there would be
solutions for
, and so on.
The total number of solutions to is
. Using the hockey stick theorem, we see this equals
, and when we take the remainder of that number when divided by
, we get the answer,
~aragornmf
Solution 2 (Official MAA)
For any nonnegative integer , the function
increases on the interval
, with
and
for every
in this interval. On this interval
, which takes on every real value in the interval
exactly once. Thus for each nonnegative real number
, the equation
has exactly one solution
for every
.
For each integer there is exactly one
with
such that
; likewise for each integer
there is exactly one
with
and
such that
. Finally, for each integer
there is exactly one
with
,
, and
such that
.
Thus has exactly one solution
with
for each triple of integers
with
, noting that
is not a solution. This nondecreasing ordered triple can be identified with a multiset of three elements of the set of
integers
, which can be selected in
ways. Thus
Video Solution
https://youtu.be/bz5N-jI2e0U?t=515
See Also
2020 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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