HMMT 二月 2006 · 冲刺赛 · 第 26 题

HMMT February 2006 — Guts Round — Problem 26

[9] Let a ≥ b ≥ c be real numbers such that 2 2 2 a bc + ab c + abc + 8 = a + b + c 2 2 2 2 2 2 a b + a c + b c + b a + c a + c b + 3 abc = − 4 2 2 2 2 2 2 a b c + ab c + a bc = 2 + ab + bc + ca 5 If a + b + c > 0, then compute the integer nearest to a .

解析

Let a ≥ b ≥ c be real numbers such that 2 2 2 a bc + ab c + abc + 8 = a + b + c 2 2 2 2 2 2 a b + a c + b c + b a + c a + c b + 3 abc = − 4 2 2 2 2 2 2 a b c + ab c + a bc = 2 + ab + bc + ca 5 If a + b + c > 0, then compute the integer nearest to a . Answer: 1279 Solution: We factor the first and third givens, obtaining the system 2 2 2 a bc + ab c + abc − a − b − c = ( abc − 1)( a + b + c ) = − 8 2 2 2 2 2 2 a b + a c + b c + b a + c a + c b + 3 abc = ( ab + bc + ca )( a + b + c ) = − 4 2 2 2 2 2 2 a b c + ab c + a bc − ab − bc − ca = ( abc − 1)( ab + bc + ca ) = 2 8 Writing X = a + b + c, Y = ab + bc + ca, Z = abc − 1, we have XZ = − 8 , XY = − 4 , Y Z = 2