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HMMT 二月 2010 · 冲刺赛 · 第 24 题

HMMT February 2010 — Guts Round — Problem 24

专题
Contest Math / 竞赛数学
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
    let a = a − a . As n tends to infinity, what is the limit of the sum of the roots of a ?5
    n +2 n +1 n n
    13 ANNUAL HARVARD-MIT MATHEMATICS TOURNAMENT, 20 FEBRUARY 2010 — GUTS ROUNDth

英文原题

[ 12 ] Define a sequence of polynomials as follows: let a 1 = 3 x 2 − x , let a 2 = 3 x 2 − 7 x + 3, and for n ≥ 1,
let a n +2 = 5
2 a n +1 − a n . As n tends to infinity, what is the limit of the sum of the roots of a n ?
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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13 th ANNUAL HARVARD-MIT MATHEMATICS TOURNAMENT, 20 FEBRUARY 2010 — GUTS ROUND

解析

英文解析

  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
    let a = a − a . As n tends to infinity, what is the limit of the sum of the roots of a ?5
    n +2 n +1 n n
    2132
    Answer: By using standard methods for solving linear recurrences , we see that this recurrence
    2 5 1 n − n 3
    has a characteristic polynomial of x − x + 1 = ( x − )( x − 2), hence a ( x ) = c ( x ) · 2 + d ( x ) · 2
    2 2 nfor some polynomials c and d . Plugging in n = 1 and n = 2 gives
    2 c ( x ) + d ( x ) = 3 x − x 21
    and 2
    4 c ( x ) + d ( x ) = 3 x − 7 x + 3 .21
    Subtracting the first equation from two times the second equation gives 6 c ( x ) = 3 x − 13 x + 6, so 24
    3 x − 13 x +62
    n − nc ( x ) = . As n grows large, the c ( x )2 term dominates compared to the d ( x )2 term, sothe roots of a ( x ) converge to the roots of c ( x ). Thus the roots of a ( x ) converge to the roots of 6
    n n
    2 313
    3 x − 13 x + 6, which by Vieta’s formula have a sum of .
    13 ANNUAL HARVARD-MIT MATHEMATICS TOURNAMENT, 20 FEBRUARY 2010 — GUTS ROUNDth