Problem:
Rodrigo has a very large piece of graph paper. First, he draws a line segment connecting point to point and colors the cells whose interiors intersect the segment, as shown below. Next, Rodrigo draws a line segment connecting point to point . Again, he colors the cells whose interiors intersect the segment. How many cells will he color this time?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
From the given example, observe that if we consider a different line segment with the same slope, for instance the line connecting points and , then the number of colored cells will be halved.
In general, it is possible to scale down the problem as long as we still have the same slope for the line. Next, note that the line segment passing through points and has a slope of . Hence, a scaled down version that we can consider is a segment connecting the points and .
From the diagram, it is evident that the line passes through cells. We know that this will happen times as the segment passes through points and . Hence, Rodrigo will need to color
Thus, is the correct answer.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions