2016 AIME I Problems/Problem 13
Notice that we don't really care about what the -coordinate of the frog is. So let's let
denote the expected number of times Freddy will jump at a
coordinate of
until he reaches the line
. So therefore we want to find
.
So we have . Suppose Freddy is at
. He has a
probability of moving horizontally,
chance of moving up and
chance of moving down. So we have
So we get the recursion
. Rearranging we see
. That means the difference between consecutive terms goes down by
each time. So for convenience let's let
and
. So that means
Yes, this is a quadratic. Now, notice that since there is a boundary, we have to give special care to
. We have
so this turns into
and this results in
. So now we know
Now, we also have that
so that gives us
so
. So now we know
so plugging in
we get