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
Because both spheres have their centers on the x-axis, we can simplify the graph a bit by looking at a two-dimensional plane (the previous z-axis is the new x-axis and the y-axis remains the same).
The spheres now become circles with centers at and . They have radii and , respectively. Let circle be the circle centered on and circle be the one centered on .
The point on circle closest to the center of circle is . The point on circle B closest to the center of circle is .
Taking a look back at the 3-dimensional coordinate grid with the spheres, we can see that their intersection appears to be a circle with congruent dome shapes on either end. All
See also
1996 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 26 |
Followed by Problem 28 | |
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