HMMT 十一月 2019 · 冲刺赛 · 第 20 题

HMMT November 2019 — Guts Round — Problem 20

[11] Consider an equilateral triangle T of side length 12. Matthew cuts T into N smaller equilateral triangles, each of which has side length 1, 3, or 8. Compute the minimum possible value of N .

解析

[11] Consider an equilateral triangle T of side length 12. Matthew cuts T into N smaller equilateral triangles, each of which has side length 1, 3, or 8. Compute the minimum possible value of N . Proposed by: Matthew Cho Answer: 16 Matthew can cut T into 16 equilateral triangles with side length 3. If he instead included a triangle 2 2 2 of side 8, then let him include a triangles of side length 3. He must include 12 − 8 − 3 a = 80 − 9 a triangles of side length 1. Thus a ≤ 8, giving that he includes at least (80 − 9 a ) + ( a ) + 1 = 81 − 8 a ≥ 17 total triangles, so 16 is minimal.