1996 AMC 12 Problem 27
Attempt Problem 27 of the 1996 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1996 AMC 12 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
27.
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?
Answer: D
Small Hint:
First intersect the possible integer ranges for the -coordinate in the two balls
Big Hint:
At the only possible height, reduce both sphere inequalities to a bound on
Solution:
The first ball permits integer heights through while the second permits through so an intersection lattice point must have At that height the two bounds are Thus can be or These give and ordered integer pairs, respectively, for points. The correct answer is D.
Problem 27 in Other Years
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