1986 AMC 8 Problem 23
Attempt Problem 23 of the 1986 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 1986 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
23.
The large circle has diameter The two small circles have their centers on and just touch at the center of the large circle. If each small circle has radius what is the value of the ratio of the area of the shaded region to the area of one of the small circles?
between and
between and
between and
cannot be determined from the information given
Answer: B
Small hint:
The large circle has radius since equals two small diameters
Big hint:
By symmetry the shaded region is half the difference between the large circle’s area and the two small circles’ area
Solution:
The large circle has radius so its area is and each small circle has area
By symmetry the shaded region is half the difference of the areas: The ratio of this to one small circle’s area is
Thus, the correct answer is B .
Problem 23 in other years
- 1985 AMC 8
- 1987 AMC 8
- 1988 AMC 8
- 1989 AMC 8
- 1990 AMC 8
- 1991 AMC 8
- 1992 AMC 8
- 1993 AMC 8
- 1994 AMC 8
- 1995 AMC 8
- 1996 AMC 8
- 1997 AMC 8
- 1998 AMC 8
- 1999 AMC 8
- 2000 AMC 8
- 2001 AMC 8
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