PUMaC 2020 · 代数（B 组） · 第 1 题

PUMaC 2020 — Algebra (Division B) — Problem 1

The function f ( x ) = x + (2 a + 3) x + ( a + 1) only has real zeroes. Suppose the smallest p possible value of a can be written in the form , where p, q are relatively prime integers. Find q | p | + | q | .

解析

The function f ( x ) = x + (2 a + 3) x + ( a + 1) only has real zeroes. Suppose the smallest p possible value of a can be written in the form , where p, q are relatively prime integers. Find q | p | + | q | . Proposed by: Frank Lu Answer: 17 Notice that for f ( x ) to have real zeroes, the discriminant needs to be nonnegative. This 2 2 2 2 requires, in turn, that (2 a + 3) − 4( a + 1) = 4 a + 12 a + 9 − 4 a − 4 = 12 a + 5 . But this − 5 requires, that a ≥ . Our answer is hence 17 . 12