HMMT 二月 2005 · 几何 · 第 6 题

英文解析

6. A triangular piece of paper of area 1 is folded along a line parallel to one of the sidesand 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 distancefrom A to BC be h , and suppose we fold along a line at a distance of ch from A . Wewill assume that neither angle B nor C is obtuse, for the area of overlap will only besmaller if either is obtuse. If c ≤ , then A does not fold past the edge BC , so the 1
   overlap is a triangle similar to the original with height ch ; the area of the figure is 2
   2 3 1
   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 removinga triangle of height (2 c − 1) h similar to the original. The area of the resulting figure is
   2 2 22
   thus 1 − c + (2 c − 1) = 3 c − 4 c + 2. This is minimized when c = , when the area
   2 3 23
   is < ; the minimum possible area is therefore .
   3 4 3