HMMT 二月 2024 · ALGNT 赛 · 第 1 题

HMMT February 2024 — ALGNT Round — Problem 1

Suppose r , s , and t are nonzero reals such that the polynomial x + rx + s has s and t as roots, and 2 the polynomial x + tx + r has 5 as a root. Compute s .

解析

Suppose r , s , and t are nonzero reals such that the polynomial x + rx + s has s and t as roots, and 2 the polynomial x + tx + r has 5 as a root. Compute s . Proposed by: Rishabh Das Answer: 29 2 Solution: The first equation implies st = s , so t = 1. Then x + x + r has 5 as a root, so r + 30 = 0, 2 implying r = − 30. Finally, x − 30 x + s has 1 as a root, so s = 29 . 2 Remark: We missed the case of s = t , so x + rx + s has s = t as one root, and 1 as the other root (by Vieta’s). This means r = − s − 1. Then 2 2 x + tx + r = x + sx − ( s + 1) = ( x + ( s + 1))( x − 1) has 5 as a root, so s = − 6 is another solution. During the competition, both the answers − 6 and 29 (as well as “29 or − 6”) were accepted.