Question

Given points $P_1,$ $P_2,$ $\ldots,$ $P_7$ on a straight line, in the order stated (not necessarily evenly spaced). Let $P$ be an arbitrarily selected point on the line and let $s$ be the sum of the undirected lengths

$PP_1,\ PP_2,\ \ldots,\ PP_7.$

Then $s$ is smallest if and only if the point $P$ is:

midway between $P_1$ and $P_7$

midway between $P_2$ and $P_6$

midway between $P_3$ and $P_5$

at $P_4$

at $P_1$

Answer: D

Concepts:optimization pairing and grouping

Difficulty rating: 1640

Small Hint:

Pair the distances to $P_1,P_7$ , then $P_2,P_6$ , then $P_3,P_5$

Big Hint:

Each paired sum is minimized throughout its intervening segment; the unpaired middle point selects one location

Solution:

For any pair $P_i,P_{8-i},$ the sum $PP_i+PP_{8-i}$ is minimized when $P$ lies between them. The three such intervals all contain $P_4.$ The remaining term $PP_4$ is uniquely minimized at $P=P_4.$

Thus, the correct answer is D.

1963 AMC 12 Problem 37

Answer: D

Difficulty rating: 1640

Small Hint:

Big Hint:

Solution:

Problem 37 in Other Years