2021 JMC 10 Problems/Problem 12
Problem
Mihir draws line and Nathan draws line
for an integer
The two lines divide the region
into four regions, with regions possibly having infinite area. What is the sum of all possible values of
Solution
The line intersects the curve
at two points, namely
and
Note that the line
must intersect
at a point strictly between the points
and
to divide
into four regions. If
intersects
at
we have
Similarly, if
intersects
at
we have
So,
The sum of all possible
is equal to