HMMT 二月 2012 · 冲刺赛 · 第 15 题
HMMT February 2012 — Guts Round — Problem 15
题目详情
英文原题
[7] Let f(x) =x2+ax+bandg(x) =x2+cx+dbe two distinct real polynomials such that the x-coordinate of the vertex of fis a root of g, thex-coordinate of the vertex of gis a root of fand both fandghave the same minimum value. If the graphs of the two polynomials intersect at the point (2012 ,−2012), what is the value of a+c?
解析
英文解析
- [ 7 ] Let f ( x ) = x + ax + b and g ( x ) = x + cx + d be two distinct real polynomials such that thex -coordinate of the vertex of f is a root of g , the x -coordinate of the vertex of g is a root of f and bothf and g have the same minimum value. If the graphs of the two polynomials intersect at the point
(2012 , − 2012), what is the value of a + c ?
Answer: − 8048 It is clear, by symmetry, that 2012 is the equidistant from the vertices of thetwo quadratics. Then it is clear that reflecting f about the line x = 2012 yields g and vice versa.
Thus the average of each pair of roots is 2012. Thus the sum of the four roots of f and g is 8048, soa + c = − 8048.