1978 AMC 12 Problem 16
Attempt Problem 16 of the 1978 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 1978 AMC 12 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
16.
In a room containing people, at least one person has not shaken hands with everyone else in the room. What is the maximum number of people in the room that could have shaken hands with everyone else?
none of these
Answer: E
Small hint:
A missed handshake always involves two people
Big hint:
Find an upper bound, then realize it by omitting just one handshake
Solution:
If one person has missed a handshake, the other person in that missed pair also has not shaken hands with everyone. Thus at most people can have shaken hands with everyone. This is attainable when exactly two people fail to shake hands with each other and every other handshake occurs. Since is not listed, the answer is “none of these.”
Therefore, the correct answer is E.
Problem 16 in other years
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