Problem:
Consider two solid spherical balls, one centered at (0,0,221) with radius 6, and the other centered at (0,0,1) with radius 29. How many points (x,y,z) with only integer coordinates (lattice points) are there in the intersection of the balls?
Answer Choices:
A. 7
B. 9
C. 11
D. 13
E. 15
Solution:
From the description of the first ball we find that z≥9/2, and from that of the second, z≤11/2. Because z must be an integer, the only possible lattice points in the intersection are of the form (x,y,5). Substitute z=5 into the inequalities defining the balls:
x2+y2+(z−221)2≤62 and x2+y2+(z−1)2≤(29)2
These yield
x2+y2+(−211)2≤62 and x2+y2+(4)2≤(29)2
which reduce to
x2+y2≤423 and x2+y2≤417
If (x,y,5) satisfies the second inequality, then it must satisfy the first one. The only remaining task is to count the lattice points that satisfy the second inequality. There are 13:
(−2,0,5),(1,−1,5),(0,−1,5),(2,0,5),(−1,1,5),(0,1,5),(0,−2,5),(1,1,5), and (0,2,5),(−1,0,5),(0,0,5).(−1,−1,5),(1,0,5),