Mock AIME 2 2010 Problems/Problem 7
As the functions map 1,2,3,4,5 to itself,, the condition reduces to f(x)=x, which counts as 1 function.
Few Quick SideNotes:
This problem actually also appeared on the 2011 HMMT and the 2018 AIME II.
However, here is the general approach on how to solve it for those interested and for questions similar to this one. :)
First of all, if we let in the equation
, we get that
.
Therefore, there must be at least one such that
. So we now do casework on the number of values of
are such that
. :)
Case 1: One Value Of u for which that is true
Then there are 5 ways to choose one number for from
So let's assume
and then multiply by
at the very end of this case. Then
for
Now we take casework on the number of numbers in the set
for which
.
If there is only one number from that set such that , there are only 4 ways to choose that number (so just assume we pick 2 to be that number), then since
then
since
is the only other value of
for which
, so this produces
cases as such.
If there are numbers from the set
such that
, then there are
ways to choose those
numbers. (This time we will assume that we pick
, and multiply by
at the end.) Then since
and
are the only other values of
besides
such that
, we know that
and
cannot equal
, so
and
can each be either
So the total number of cases for this subcase is
.
If there are numbers from the set
such that
, then there are
ways to choose those
numbers. (This time we will assume that we pick
, and multiply by
at the end.) Then since
are the only other values of
besides
such that
,
, so
can either be
or
. This gives us
cases.
If there are numbers from the set
such that
, then there is only 1 way to do that. XD
To be honest, the rest of the cases should be pretty similar to the first case, so I will let the aops users who visit this problem's solution think about it themselves and connect the dots for the remaining cases. :)
Case 2: Two Values Of u for which that is true
Case 3: Three Values Of u for which that is true
Case 4: Four Values Of u for which that is true
Case 5: Five Values Of u for which that is true
Hope this helps! :)
~Professor-Mom