Problem:
Let be the set of lattice points in the coordinate plane, both of whose coordinates are integers between 1 and 30 , inclusive. Exactly 300 points in lie on or below a line with equation . The possible values of lie in an interval of length , where and are relatively prime positive integers. What is ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Answer (E): Because the triangle bounded by the -axis and the lines and has area , it is reasonable to guess that the acceptable values of are near . To count the number of elements of that are on or below the line , let be the number of such points with for . Note that , and for . Thus the number of elements of that are on or below the line is . Because some lattice points in do lie on the line, is the least possible value of .
For , the line is of a unit below the point of a unit below the point , and 1 unit below the point . Therefore as increases from , the first lattice point to be intersected by the line is . Hence the possible values of satisfy , so they lie in an interval of length . The requested sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions