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 (Problems • Answer Key • Resources) Preceded byProblem 36 Followed byProblem 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.