1996 AHSME Problems/Problem 27
Contents
Problem
Consider two solid spherical balls, one centered at with radius , and the other centered at with radius . How many points with only integer coordinates (lattice points) are there in the intersection of the balls?
Solution 1
The two equations of the balls are
Note that along the axis, the first ball goes from , and the second ball goes from . The only integer value that can be is .
Plugging that in to both equations, we get:
The second inequality implies the first inequality, so the only condition that matters is the second inequality.
From here, we do casework, noting that :
For , we must have . This gives points.
For , we can have . This gives points.
For , we can have . This gives points.
Thus, there are possible points, giving answer .
Solution 2
See also
1996 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 26 |
Followed by Problem 28 | |
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