HMMT 二月 2005 · GEN1 赛 · 第 9 题

HMMT February 2005 — GEN1 Round — Problem 9

A triangular piece of paper of area 1 is folded along a line parallel to one of the sides and pressed flat. What is the minimum possible area of the resulting figure? 2

解析

A triangular piece of paper of area 1 is folded along a line parallel to one of the sides and pressed flat. What is the minimum possible area of the resulting figure? Solution: 2/3 Let the triangle be denoted ABC , and suppose we fold parallel to BC . Let the distance from A to BC be h , and suppose we fold along a line at a distance of ch from A . We will assume that neither angle B nor C is obtuse, for the area of overlap will only be 1 smaller if either is obtuse. If c ≤ , then A does not fold past the edge BC , so the 2 overlap is a triangle similar to the original with height ch ; the area of the figure is 3 1 2 then 1 − c ≥ . Suppose c > , so that A does fold past BC . Then the overlap is a 4 2 trapezoid formed by taking a triangle of height ch similar to the original and removing a triangle of height (2 c − 1) h similar to the original. The area of the resulting figure is 2 2 2 2 thus 1 − c + (2 c − 1) = 3 c − 4 c + 2. This is minimized when c = , when the area 3 2 3 2 is < ; the minimum possible area is therefore . 3 4 3 2