HMMT 十一月 2023 · THM 赛 · 第 6 题

HMMT November 2023 — THM Round — Problem 6

A function g is ever more than a function h if, for all real numbers x , we have g ( x ) ≥ h ( x ) . Consider 2 2 all quadratic functions f ( x ) such that f (1) = 16 and f ( x ) is ever more than both ( x + 3) and x + 9 . Across all such quadratic functions f , compute the minimum value of f (0) .

解析

A function g is ever more than a function h if, for all real numbers x , we have g ( x ) ≥ h ( x ). Consider 2 2 all quadratic functions f ( x ) such that f (1) = 16 and f ( x ) is ever more than both ( x + 3) and x + 9. Across all such quadratic functions f , compute the minimum value of f (0). Proposed by: Isabella Quan, Pitchayut Saengrungkongka, Alex Yi 21 Answer: 2 2 2 Solution: Let g ( x ) = ( x + 3) and h ( x ) = x + 9. Then f (1) = g (1) = 16. Thus, f ( x ) − g ( x ) has a root at x = 1. Since f is ever more than g , this means that in fact 2 f ( x ) − g ( x ) = c ( x − 1) for some constant c . Now 2 2 f ( x ) − h ( x ) = (( f ( x ) − g ( x )) + ( g ( x ) − h ( x )) = c ( x − 1) + 6 x = cx − (2 c − 6) x + c is always nonnegative. The discriminant is 2 2 (2 c − 6) − 4 c = 24 c − 36 ≥ 0 , 3 so the smallest possible value of c is . Then 2 21 2 f (0) = g (0) + c ( x − 1) = 9 + c ≥ , 2 3 with equality at c = . 2