2011 AMC 10A Problems/Problem 25
Problem 25
Let be a square region and
an integer. A point
in the interior of
is called
partitional if there are
rays emanating from
that divide
into
triangles of equal area. How many points are 100-ray partitional but not 60-ray partitional?
Solution
The domain of is defined when
.
. Applying the domain of
and the fact that square roots must be positive, we get
. Simplify this to arrive at the domain for
, which is defined when
. Repeat this process for
to get a domain of
. For
, since square roots are positive, we can exclude the negative values of the previous domain to arrive at
as the domain of
. We now arrive at a domain with a single number that defines
, however, since we are looking for the largest value for
for which the domain of
is nonempty, we must continue until we arrive at a domain that is empty. We continue with
to get a domain of
. Solve for
to get
. Since square roots cannot be negative, this is the last nonempty domain. We add to get
.