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HMMT 二月 2010 · 冲刺赛 · 第 24 题

HMMT February 2010 — Guts Round — Problem 24

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 12 ] Define a sequence of polynomials as follows: let a = 3 x − x , let a = 3 x − 7 x + 3, and for n ≥ 1, 1 2 5 let a = a − a . As n tends to infinity, what is the limit of the sum of the roots of a ? n +2 n +1 n n 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th 13 ANNUAL HARVARD-MIT MATHEMATICS TOURNAMENT, 20 FEBRUARY 2010 — GUTS ROUND
解析
  1. [ 12 ] Define a sequence of polynomials as follows: let a = 3 x − x , let a = 3 x − 7 x + 3, and for n ≥ 1, 1 2 5 let a = a − a . As n tends to infinity, what is the limit of the sum of the roots of a ? n +2 n +1 n n 2 13 2 Answer: By using standard methods for solving linear recurrences , we see that this recurrence 3 2 5 1 n − n has a characteristic polynomial of x − x + 1 = ( x − )( x − 2), hence a ( x ) = c ( x ) · 2 + d ( x ) · 2 n 2 2 for some polynomials c and d . Plugging in n = 1 and n = 2 gives 1 2 2 c ( x ) + d ( x ) = 3 x − x 2 and 1 2 4 c ( x ) + d ( x ) = 3 x − 7 x + 3 . 4 2 Subtracting the first equation from two times the second equation gives 6 c ( x ) = 3 x − 13 x + 6, so 2 3 x − 13 x +6 n − n c ( x ) = . As n grows large, the c ( x )2 term dominates compared to the d ( x )2 term, so 6 the roots of a ( x ) converge to the roots of c ( x ). Thus the roots of a ( x ) converge to the roots of n n 13 2 3 3 x − 13 x + 6, which by Vieta’s formula have a sum of . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th 13 ANNUAL HARVARD-MIT MATHEMATICS TOURNAMENT, 20 FEBRUARY 2010 — GUTS ROUND