HMMT 二月 2025 · 冲刺赛 · 第 22 题

HMMT February 2025 — Guts Round — Problem 22

[12] Let a , b , and c be real numbers such that a ( b + c ) = 1, b ( c + a ) = 2, and c ( a + b ) = 5. Given that there are three possible values for abc , compute the minimum possible value of abc .

解析

[12] Let a , b , and c be real numbers such that a ( b + c ) = 1, b ( c + a ) = 2, and c ( a + b ) = 5. Given that there are three possible values for abc , compute the minimum possible value of abc . Proposed by: Pitchayut Saengrungkongka √ − 5 − 5 Answer: 2 Solution: Let x = abc . Multiplying all equations together and simplifying gives 2 ( abc ) ( a + b )( b + c )( c + a ) = 10 , 2 2 2 2 ( abc ) a ( b + c ) + b ( c + a ) + c ( a + b ) + 2 abc = 10 , 2 x (1 + 2 + 5 + 2 x ) = 10 , 2 x ( x + 4) = 5 . 2 The resulting cubic factors as ( x − 1)( x + 5 x + 5) = 0. Therefore, the smallest possible value of abc √ √ 2 − 5 − 5 − 4 · 5 − 5 − 5 is = . 2 2