1986 AMC 8 Problem 22
Attempt Problem 22 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).
22.
Alan, Beth, Carlos, and Diana were discussing their possible grades in mathematics class this grading period. Alan said, “If I get an A, then Beth will get an A.” Beth said, “If I get an A, then Carlos will get an A.” Carlos said, “If I get an A, then Diana will get an A.” All of these statements were true, but only two of the students received an A. Which two received A’s?
Alan, Beth
Beth, Carlos
Carlos, Diana
Alan, Diana
Beth, Diana
Answer: C
Small hint:
If a student earlier in the chain gets an A, everyone after them in the chain must also get an A
Big hint:
For exactly two A’s, the two must be the last students in the chain
Solution:
The statements form a chain: Alan’s A forces Beth’s, Beth’s forces Carlos’s, and Carlos’s forces Diana’s. If Alan got an A, all four would; if Beth got an A, three would.
The only way to have exactly two A’s is for them to be the last two in the chain, Carlos and Diana.
Thus, the correct answer is C .
Problem 22 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
- 2002 AMC 8
- 2003 AMC 8
- 2004 AMC 8
- 2005 AMC 8
- 2006 AMC 8
- 2007 AMC 8
- 2008 AMC 8
- 2009 AMC 8
- 2010 AMC 8
- 2011 AMC 8
- 2012 AMC 8
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