Problem:
Let be the set of points whose coordinates , and are integers that satisfy , and . Two distinct points are randomly chosen from . The probability that the midpoint of the segment they determine also belongs to is , where and are relatively prime positive integers. Find .
Solution:
Because the points of have integer coordinates, they are called lattice points. There are ways to choose a first lattice point and then a distinct second. In order for their midpoint to be a lattice point, it is necessary and sufficient that corresponding coordinates have the same parity. There are ways for the first coordinates to have the same parity, including ways in which the coordinates are the same. There are ways for the second coordinates to have that same parity, including ways in which the coordinates are the same. There are ways for the third coordinates to have the same parity, including in which the coordinates are the same. It follows that there are ways to choose two distinct lattice points, so that the midpoint of the resulting segment is also a lattice point. The requested probability is , so .
Because there are points to choose from, there are ways to choose the two points. In order that the midpoint of the segment joining the two chosen points also be a lattice point, it is necessary and sufficient that corresponding coordinates have the same parity. Notice that there are
points whose coordinates are all even,
points whose coordinates are all odd,
points whose only odd coordinate is ,
points whose only odd coordinate is ,
points whose only odd coordinate is ,
points whose only even coordinate is ,
points whose only even coordinate is , and
points whose only even coordinate is .
Thus the desired number of segments is
so that the requested probability is .
The problems on this page are the property of the MAA's American Mathematics Competitions