HMMT 十一月 2012 · THM 赛 · 第 6 题

HMMT November 2012 — THM Round — Problem 6

[ 3 ] A rectangular piece of paper with vertices ABCD is being cut by a pair of scissors. The pair of scissors starts at vertex A , and then cuts along the angle bisector of DAB until it reaches another edge of the paper. One of the two resulting pieces of paper has 4 times the area of the other piece. What is the ratio of the longer side of the original paper to the shorter side?

解析

[ 3 ] A rectangular piece of paper with vertices ABCD is being cut by a pair of scissors. The pair of scissors starts at vertex A , and then cuts along the angle bisector of DAB until it reaches another edge of the paper. One of the two resulting pieces of paper has 4 times the area of the other piece. What is the ratio of the longer side of the original paper to the shorter side? 5 Answer: Without loss of generality, let AB > AD , and let x = AD , y = AB . Let the cut along 2 the angle bisector of ∠ DAB meet CD at E . Note that ADE is a 45-45-90 triangle, so DE = AD = x , 2 x x and EC = y − x . Now, [ ADE ] = , and [ AECB ] = x ( y − ) = 4[ ADE ]. Equating and dividing both 2 2 x 5 sides by x , we find that 2 x = y − , so y/x = . 2 2