1963 AHSME Problems/Problem 37

Problem

Given points $P_1, P_2,\cdots,P_7$ 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 $PP_1, PP_2, \cdots , PP_7$. Then $s$ is smallest if and only if the point $P$ is:

$\textbf{(A)}\ \text{midway between }P_1\text{ and }P_7\qquad \\ \textbf{(B)}\ \text{midway between }P_2\text{ and }P_6\qquad \\ \textbf{(C)}\ \text{midway between }P_3\text{ and }P_5\qquad \\ \textbf{(D)}\ \text{at }P_4 \qquad \textbf{(E)}\ \text{at }P_1$

Solution

By the Triangle Inequality, $P_1P + P_7P \ge P_1P_7$, with equality happening when $P$ is between $P_1$ and $P_7$. Using similar logic, $P$ must be between $P_3$ and $P_5$ in order for the distance to be minimized.

The only point left to deal with is $P_4$ (which is also between $P_3$ and $P_5$). The minimum possible distance is $0$ (when $P$ is on $P_4$), so the answer is $\boxed{\textbf{(D)}}$.

See Also

1963 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 36
Followed by
Problem 38
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
All AHSME Problems and Solutions


The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Invalid username
Login to AoPS