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, 2022

Geometry Difficulty 4.8 AIME Prove it United States

Problem:
Let $ABC$ be a triangle with $AB = 2021$ , $AC = 2022$ , and $BC = 2023$ . Compute the minimum value of $AP + 2BP + 3CP$ over all points $P$ in the plane.

Solutions — 2

Solution 1

Solution:
The minimizing point is when $P = C$ . To prove this, consider placing $P$ at any other point $O \neq C$ . Then, by moving $P$ from $O$ to $C$ , the expression changes by
$(AC - AO) + 2(BC - BO) + 3(CC - CO) < OC + 2OC - 3OC = 0$
by the triangle inequality. Since this is negative, $P = C$ must be the optimal point. The answer is $2022 + 2 \cdot 2023 + 3 \cdot 0 = 6068$ .

Solution 2

Solution:
We use a physical interpretation. Imagine an object acted upon by forces of magnitudes $1$ , $2$ , and $3$ towards $A$ , $B$ , and $C$ , respectively. The potential energy of the object at point $P$ in this system is $AP + 2BP + 3CP$ . This potential energy is minimized when the object experiences $0$ net force; in this case, it occurs when it is exactly at point $C$ (because the pull towards $C$ overpowers the other two forces combined).

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