Problem:
The graph of the equation is drawn on graph paper with each square representing one unit in each direction. How many of the by graph paper squares have interiors lying entirely below the graph and entirely in the first quadrant?
Solution:
The graph passes through the points , where each decrease of 1 unit in results in an increase of units in . Therefore the region can be divided into rectangles with dimensions , , and , for a total of squares.
Consider the rectangle with vertices , and . There are by squares within this rectangle. The diagonal from to crosses exactly one of these squares between and for most of the possible values of . There are exactly values of for which the diagonal crosses one of the horizontal lines , and for these values the diagonal crosses two squares. The diagonal never passes through any corners, because and are relatively prime and . Thus, out of the squares, of them are crossed by the diagonal, leaving squares untouched. Half of these, or of them, lie below the diagonal.
The problems on this page are the property of the MAA's American Mathematics Competitions