1982 AMC 12 Problem 29
Attempt Problem 29 of the 1982 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 1982 AMC 12 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
29.
Let and be three positive real numbers whose sum is If no one of these numbers is more than twice any other, then the minimum possible value of the product is
none of these
Answer: A
Small hint:
Order the variables ; then the active constraint is
Big hint:
A minimum occurs on the boundary where
Solution:
Order At a minimum the spread is maximal, so and The ordering requires Thus Its only interior critical point is a maximum, so compare endpoints: the values are and respectively. The minimum is
Therefore, the correct answer is A.
Problem 29 in other years
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