1963 AMC 12 Problem 37
Attempt Problem 37 of the 1963 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 1963 AMC 12 solutions, or check the answer key.
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37.
Given points on a straight line, in the order stated (not necessarily evenly spaced). Let be an arbitrarily selected point on the line and let be the sum of the undirected lengths
Then is smallest if and only if the point is:
midway between and
midway between and
midway between and
at
at
Answer: D
Small hint:
Pair the distances to , then , then
Big hint:
Each paired sum is minimized throughout its intervening segment; the unpaired middle point selects one location
Solution:
For any pair the sum is minimized when lies between them. The three such intervals all contain The remaining term is uniquely minimized at
Thus, the correct answer is D.