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),​