2022 AMC 12A Problems/Problem 25
Problem
A circle with integer radius is centered at
. Distinct line segments of length
connect points
to
for
and are tangent to the circle, where
,
, and
are all positive integers and
. What is the ratio
for the least possible value of
?
Solution
Case 1: The tangent and the origin are on the opposite sides of the circle.
In this case, .
We can easily prove that
Recall that .
Taking square of (1) and reorganizing all terms, (1) is converted as
Case 2: The tangent and the origin are on the opposite sides of the circle.
In this case, .
We can easily prove that
Recall that .
Taking square of (2) and reorganizing all terms, (2) is converted as
Putting both cases together, for given , we look for solutions of
and
satisfying
with either
or
.
Now, we need to find the smallest , such that the number of feasible solutions of
is at least 14.
For equation
we observe that the R.H.S. is a not a perfect square. Thus, the number of positive
is equal to the number of positive divisors of
.
Second, for each feasible positive solution , its opposite
is also a solution. However,
corresponds to a feasible solution if
with
and
, but
may not lead to a feasible solution if
with
and
.
Recall that we are looking for that leads to at least 14 solutions.
Therefore, the above observations imply that we must have
, such that
has least 7 positive divisors.
Following this guidance, we find the smallest is 6. This leads to the following solutions:
,
.
,
.
,
.
,
.
,
.
,
.
,
.
Therefore, .
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)