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HMMT 二月 2014 · 冲刺赛 · 第 26 题

HMMT February 2014 — Guts Round — Problem 26

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 17 ] For 1  j  2014, define
    2014
    2014 2014 2014Yb = j ( i j )
    i =1 ,i 6 = jjwhere the product is over all i 2 { 1 , . . . , 2014 } except i = j . Evaluate
    1 1 1 + + · · · + .
    b b b
    1 2 2014
解析

英文解析

  1. [ 17 ] For 1 ≤ j ≤ 2014, define
    2014
    ∏
    2014 2014 2014
    b = j ( i − j )
    i =1 ,i 6 = jjwhere the product is over all i ∈ { 1 , . . . , 2014 } except i = j . Evaluate
    1 1 1 + + · · · + .
    b b b
    1 2 2014
    Answer: We perform Lagrange interpolation on the polynomial P ( x ) = 1 through the 1
    2014
    2014!
    2014 2014 2014
    points 1 , 2 , . . . , 2014 . We have
    ∏
    2014
    2014
    2014
    ∑
    ( x − i )
    i =1 ,i 6 = j
    1 = P ( x ) = .
    ∏
    2014
    2014 2014
    ( j − i )
    j =1 i =1 ,i 6 = j
    Thus,
    2014
    2014 2013 2014!
    ∑
    (( − 1) )
    2014
    1 = P (0) = ,j
    ∏
    2014
    2013 2014 2014
    ( − 1) ( i − j )
    i =1 ,i 6 = jj =1
    which equals
    ( )
    2014
    ∑
    1 1 1 1
    2014 2014
    2014! = 2014! + + · · · + ,
    ∏
    2014
    2014 2014 2014
    b b bj ( i − j ) 1 2 2014
    i =1 ,i 6 = jj =1
    so the desired sum is .1
    2014
    2014!