HMMT 十一月 2018 · 冲刺赛 · 第 14 题

HMMT November 2018 — Guts Round — Problem 14

[ 9 ] Call a triangle nice if the plane can be tiled using congruent copies of this triangle so that any two triangles that share an edge (or part of an edge) are reflections of each other via the shared edge. How many dissimilar nice triangles are there?

解析

[ 9 ] Call a triangle nice if the plane can be tiled using congruent copies of this triangle so that any two triangles that share an edge (or part of an edge) are reflections of each other via the shared edge. How many dissimilar nice triangles are there? Proposed by: Yuan Yao Answer: 4 The triangles are 60-60-60, 45-45-90, 30-60-90, and 30-30-120. We make two observations. • By reflecting ”around” the same point, any angle of the triangle must be an integer divisor of 360 . 360 • if any angle is an odd divisor of 360 , i.e equals for odd k , then the two adjacent sides must k be equal. We do casework on the largest angle. • 60 . We are forced into a 60-60-60 triangle, which works. • 72 . By observation 2, this triangle’s other two angles are 54 . This is not an integer divisor of 360 . • 90 . The second largest angle is at least 45 . If it is 45 , it is the valid 90-45-45 triangle. If it 360 is , the triangle is invalid by observation 2. If it is 60 , it is the valid 90-60-30 triangle. If 7 it is 72 , the triangle is invalid by observation 2. • 120 . By observation 2, the other angles are 30 , meaning it is the valid 120-30-30 triangle. The conclusion follows.