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Geometry Difficulty 4.9 AIME Prove it United States

Problem:

Let $ABC$ be a triangle with $AB=13$ , $BC=14$ , $CA=15$ . Company XYZ wants to locate their base at the point $P$ in the plane minimizing the total distance to their workers, who are located at vertices $A$ , $B$ , and $C$ . There are $1$ , $5$ , and $4$ workers at $A$ , $B$ , and $C$ , respectively. Find the minimum possible total distance Company XYZ's workers have to travel to get to $P$ .

Solution

Solution:

We want to minimize $1 \cdot PA + 5 \cdot PB + 4 \cdot PC$ . By the triangle inequality, $(PA + PB) + 4(PB + PC) \geq AB + 4 BC = 13 + 56 = 69$ , with equality precisely when $P = [AB] \cap [BC] = B$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.