HMMT 二月 2012 · 冲刺赛 · 第 15 题

HMMT February 2012 — Guts Round — Problem 15

[ 7 ] Let f ( x ) = x + ax + b and g ( x ) = x + cx + d be two distinct real polynomials such that the x -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 both f 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 ?

解析

[ 7 ] Let f ( x ) = x + ax + b and g ( x ) = x + cx + d be two distinct real polynomials such that the x -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 both f 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 the two 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, so a + c = − 8048.