HMMT 十一月 2022 · 冲刺赛 · 第 23 题

HMMT November 2022 — Guts Round — Problem 23

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

解析

[12] Let ABC be a triangle with AB = 2021 , AC = 2022 , and BC = 2023. Compute the minimum value of AP + 2 BP + 3 CP over all points P in the plane. Proposed by: Evan Erickson Answer: 6068 Solution 1: The minimizing point is when P = C . To prove this, consider placing P at any other point O ̸ = C . Then, by moving P from O to C , the expression changes by ( AC − AO ) + 2( BC − BO ) + 3( CC − CO ) < OC + 2 OC − 3 OC = 0 by the triangle inequality. Since this is negative, P = C must be the optimal point. The answer is 2022 + 2 · 2023 + 3 · 0 = 6068. Solution 2: 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 + 2 BP + 3 CP . 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).