HMMT 二月 2007 · TEAM2 赛 · 第 5 题

英文解析

5. [ 20 ] The curves y = x ( x − 3) and y = ( x − 1)( x − 2) intersect at a number of points in the realplane. Determine the sum of the x -coordinates of these points of intersection.
   Answer: 7 . Because the first curve touches the x -axis at x = 0 and x = 3 while the second curvecrosses the x -axis at x = ± 1 and x = 2 , there are four points of intersection. In particular, the pointsof intersection have x -coordinates determined by the difference of the two curves:
   2 2 2 4 3 3 4 3
   0 = x ( x − 3) − ( x − 1)( x − 2) = ( x − 6 x + · · · ) − ( x + · · · ) = x − 7 x + · · · .
   ( ) − 7
   We need only the first two coefficients to determine x + x + x + x = − = 7 .
   1 2 3 4
   11