HMMT 十一月 2022 · 冲刺赛 · 第 16 题

HMMT November 2022 — Guts Round — Problem 16

[10] Given an angle θ , consider the polynomial 2 P ( x ) = sin( θ ) x + (cos( θ ) + tan( θ )) x + 1 . Given that P only has one real root, find all possible values of sin( θ ) .

解析

[10] Given an angle θ , consider the polynomial 2 P ( x ) = sin( θ ) x + (cos( θ ) + tan( θ )) x + 1 . Given that P only has one real root, find all possible values of sin( θ ) . Proposed by: Eric Shen √ 5 − 1 Answer: 0 , 2 Solution: Note that if sin( θ ) = 0, then the polynomial has 1 root. Now assume this is not the case - then the polynomial is a quadratic in x . 1 Factor the polynomial as (tan( θ ) x + 1)( x + sec( θ )). Then the condition is equivalent to sec( θ ) = , tan( θ ) √ 2 5 − 1 2 which is equivalent to sin( θ ) = cos ( θ ) = 1 − sin ( θ ). Solving now gives sin( θ ) = as the only 2 solution.