Problem:
Find the number of ordered pairs (x,y) of positive integers that satisfy x≤2y≤60 and y≤2x≤60.
Solution:
Points that have integer coordinates are called lattice points. The lattice points in question lie within the square defined by 1≤x≤30 and 1≤y≤30. The only lattice points that are not included are those for which 2y<x or 2x<y. For positive x and y, these conditions are mutually exclusive. Within the square, the inequality 2y<x cannot hold for y≥15. For each integer y between 1 and 14, inclusive, there are 30−2y such points that satisfy 2y<x, namely those for which 2y+1≤x≤30. The number of points that satisfy 2x<y is the same as the number of points that satisfy 2y<x. The total number of omitted points is therefore
2(2+4+⋯+28)=4(1+2+3+⋯+14)=420
making the answer 900−420=480.
OR
The conditions in the problem can be expressed as 1≤x≤30,y/2≤x≤2y, and 1≤y≤30. For each value of y from 1 to 15,x must be between ⌈2y⌉ and 2y, inclusive, so there are 2y−⌈2y⌉+1 values of x. (The value ⌈r⌉ of the ceiling function is the smallest integer that is not less than r.) For each value of y from 16 to 30, similar reasoning shows that there are 30−⌈2y⌉+1 values of x. The number of ordered pairs is thus
y=1∑15(2y−⌈2y⌉+1)+y=16∑30(30−⌈2y⌉+1)=(y=1∑152y)+15+15⋅31−y=1∑30⌈2y⌉=2y=1∑15y+480−2y=1∑15y=480.
The problems on this page are the property of the MAA's American Mathematics Competitions