HMMT 十一月 2023 · 冲刺赛 · 第 19 题

HMMT November 2023 — Guts Round — Problem 19

[11] Suppose a , b , and c be real numbers such that 2 a − bc = 14 , 2 b − ca = 14 , and 2 c − ab = − 3 . Compute | a + b + c | . − − →

解析

[11] Suppose a , b , and c are real numbers such that 2 a − bc = 14 , 2 b − ca = 14 , and 2 c − ab = − 3 . Compute | a + b + c | . Proposed by: Rishabh Das 17 Answer: 5 Solution: Subtracting the first two equations gives ( a − b )( a + b + c ) = 0, so either a = b or a + b + c = 0. However, subtracting first and last equations gives ( a − c )( a + b + c ) = 17, so a + b + c ̸ = 0. This means a = b . 2 17 Now adding all three equations gives ( a − c ) = 25, so a − c = ± 5. Then a + b + c = ± . 5 − − →