1980 AMC 12 Problem 26
Attempt Problem 26 of the 1980 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 1980 AMC 12 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
26.
Four balls of radius are mutually tangent, three resting on the floor and the fourth resting on the others. A tetrahedron, each of whose edges has length is circumscribed around the balls. Then equals
Answer: E
Small hint:
The four ball centers form a regular tetrahedron of edge
Big hint:
The outer faces are parallel to the corresponding center-tetrahedron faces and one radius farther away
Solution:
The ball centers form a regular tetrahedron of edge Its inradius is Each face of the circumscribed tetrahedron is parallel to the corresponding face of this center tetrahedron and lies one unit farther from the common center. By similarity, Hence
Therefore, the correct answer is E.
Problem 26 in other years
- 1950 AMC 12
- 1951 AMC 12
- 1952 AMC 12
- 1953 AMC 12
- 1954 AMC 12
- 1955 AMC 12
- 1956 AMC 12
- 1957 AMC 12
- 1958 AMC 12
- 1959 AMC 12
- 1960 AMC 12
- 1961 AMC 12
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- 1963 AMC 12
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