HMMT 十一月 2024 · 冲刺赛 · 第 20 题

HMMT November 2024 — Guts Round — Problem 20

[11] There exists a unique line tangent to the graph of y = x − 20 x + 24 x − 20 x + 25 at two distinct points. Compute the product of the x -coordinates of the two tangency points.

解析

[11] There exists a unique line tangent to the graph of y = x − 20 x + 24 x − 20 x + 25 at two distinct points. Compute the product of the x -coordinates of the two tangency points. Proposed by: Pitchayut Saengrungkongka Answer: − 38 2 Solution: If f ( x ) is tangent to the x -axis at ( c, 0), then f ( x ) will be divisible by ( x − c ) . Thus, if 2 2 f ( x ) is tangent at the x -axis at c and c , then f ( x ) = P ( x )( x − c ) ( x − c ) for some polynomial 1 2 1 2 P ( x ). By adding mx + b , we see that f ( x ) is tangent to y = mx + b at x -coordinates c and c if and 1 2 only if 2 2 f ( x ) = P ( x )( x − c ) ( x − c ) + mx + b 1 2 for some polynomial P ( x ). 4 3 2 In the case of our problem, f ( x ) = x − 20 x + 24 x − 20 x + 5, we have by comparing the leading coefficient that P ( x ) = 1. Thus, 4 2 2 2 2 x − 20 x + 24 x − 20 x + 25 = ( x − c ) ( x − c ) + mx + b. 1 2 By Vieta’s formulas, 2( c + c ) = 20 1 2 2 2 c + c + 4 c c = 24 . 1 2 1 2 Hence, 2 2 2 24 − 100 c + 2 c c + c = 10 = 100 = ⇒ c c = = − 38 , 1 2 1 2 1 2 2 which is the answer.