HMMT 二月 2023 · 冲刺赛 · 第 17 题

HMMT February 2023 — Guts Round — Problem 17

[16] An equilateral triangle lies in the Cartesian plane such that the x -coordinates of its vertices are 3 2 pairwise distinct and all satisfy the equation x − 9 x + 10 x + 5 = 0. Compute the side length of the triangle.

解析

[16] An equilateral triangle lies in the Cartesian plane such that the x -coordinates of its vertices are 3 2 pairwise distinct and all satisfy the equation x − 9 x + 10 x + 5 = 0. Compute the side length of the triangle. Proposed by: Pitchayut Saengrungkongka √ √ Answer: 68 = 2 17 Solution: Let three points be A, B , and C with x -coordinates a , b , and c , respectively. Let the ◦ circumcircle of 4 ABC meet the line y = b at point P . Then, we have ∠ BP C = 60 = ⇒ P C = 2 2 √ √ ( c − b ). Similarly, AP = ( b − a ). Thus, by the Law of Cosines, 3 3 2 2 2 ◦ AC = AP + P C − 2 · AP · P C cos 120 ( ) 4 2 2 = ( c − b ) + ( b − a ) + ( c − b )( b − a ) 3 ( ) 4 2 2 2 = a + b + c − ab − bc − ca 3 ( ) 4 2 = ( a + b + c ) − 3( ab + bc + ca ) . 3 2 4 By Vieta’s we have a + b + c = 9 and ab + bc + ca = 10, so we have AC = (81 − 30) = 68, implying 3 √ √ that the answer is 68 = 2 17.