HMMT 二月 2007 · TEAM2 赛 · 第 5 题

HMMT February 2007 — TEAM2 Round — Problem 5

[ 20 ] The curves y = x ( x − 3) and y = ( x − 1)( x − 2) intersect at a number of points in the real plane. Determine the sum of the x -coordinates of these points of intersection.

解析

[ 20 ] The curves y = x ( x − 3) and y = ( x − 1)( x − 2) intersect at a number of points in the real plane. 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 curve crosses the x -axis at x = ± 1 and x = 2 , there are four points of intersection. In particular, the points of 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 1 1