HMMT 十一月 2020 · GEN 赛 · 第 2 题

HMMT November 2020 — GEN Round — Problem 2

Let T be a trapezoid with two right angles and side lengths 4, 4, 5, and 17. Two line segments are drawn, connecting the midpoints of opposite sides of T and dividing T into 4 regions. If the difference between the areas of the largest and smallest of these regions is d , compute 240 d .

解析

Let T be a trapezoid with two right angles and side lengths 4, 4, 5, and 17. Two line segments are drawn, connecting the midpoints of opposite sides of T and dividing T into 4 regions. If the difference between the areas of the largest and smallest of these regions is d , compute 240 d . Proposed by: Shengtong Zhang Answer: 120 Solution: D C A B By checking all the possibilities, one can show that T has height 4 and base lengths 4 and 5. Orient T so that the shorter base is on the top. 4+5 9 Then, the length of the cut parallel to the bases is = . Thus, the top two pieces are trapezoids 2 2 9 with height 2 and base lengths 2 and , while the bottom two pieces are trapezoids with height 2 and 4 9 5 base lengths and . Thus, using the area formula for a trapezoid, the difference between the largest 4 2 and smallest areas is ( ) 5 9 9