PUMaC 2020 · 团队赛 · 第 5 题
PUMaC 2020 — Team Round — Problem 5
题目详情
- Suppose two polygons may be glued together at an edge if and only if corresponding edges of the same length are made to coincide. A 3 × 4 rectangle is cut into n pieces by making straight line cuts. What is the minimum value of n so that it’s possible to cut the pieces in such a way that they may be glued together two at a time into a polygon with perimeter at least 2021?
解析
- Suppose two polygons may be glued together at an edge if and only if corresponding edges of the same length are made to coincide. A 3 × 4 rectangle is cut into n pieces by making straight line cuts. What is the minimum value of n so that it’s possible to cut the pieces in such a way that they may be glued together two at a time into a polygon with perimeter at least 2021? Proposed by: Austen Mazenko Answer: 202 For n pieces, edges must be glued together at least n − 1 times, and each gluing event reduces the overall perimeter by twice the length of the edges being glued together. Furthermore, every time a cut is made to divide the bar into more pieces, it increases the total perimeter by at most twice the length of the largest cut, which is 5 (the length of the rectangle’s diagonal). To form n pieces, there are at most n − 1 cuts. Hence, an upper bound for the perimeter is 3 + 4 + 3 + 4 + 2 · 5 · ( n − 1) − 2 · 0 · ( n − 1) = 10 n + 4 since every edge being glued together has a length > 0 and all cuts have length ≤ 5. Accordingly, we need 10 n + 4 ≥ 2021 = ⇒ n ≥ 202 since n must be an integer. To see that n = 202 is sufficient, put the bar on the coordinate i plane so that it has one vertex on the origin and one at (4 , 3). First, make 200 cuts from ( , 0) N i to (4 , 3 − ) for 1 ≤ i ≤ 200 and some large integer N . N Finally, cut the bottom right triangle like so: 1 N 1 N 1 Now, all of the thin strips have two edges of length , so they may be glued together in N sequence like so: 3 √ ( ) ( ) 2 2 201 201 1 By Pythagorean Theorem, each cut has length at least 3 − + 4 − − . Making N N N N arbitrarily large, each cut may have a length sufficiently close to 5 and each small edge may have sufficiently small length so that the perimeter will exceed 2021, as desired.