1991 AMC 8 Problem 24
Attempt Problem 24 of the 1991 AMC 8 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 1991 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
24.
A cube of edge cm is cut into smaller cubes, not all the same size. If the edge of each of the smaller cubes is a whole number of centimeters, then
Answer: E
Small hint:
The smaller cubes must have whole-number edges, so each is edge or edge
Big hint:
Only one edge- cube fits; the rest are unit cubes filling the remaining volume
Solution:
The cube has volume Since the cubes are smaller and have whole-number edge lengths, their edges are or . Two edge- cubes cannot be disjoint inside the original cube: separating them would require a total span of at least in some direction. Thus at most one edge- cube fits.
The remaining of volume is filled by unit cubes. That is cubes.
Thus, the correct answer is E .
Problem 24 in other years
- 1985 AMC 8
- 1986 AMC 8
- 1987 AMC 8
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