HMMT 二月 2014 · 冲刺赛 · 第 26 题
HMMT February 2014 — Guts Round — Problem 26
题目详情
- [ 17 ] For 1 j 2014, define 2014 Y 2014 2014 2014 b = j ( i j ) j i =1 ,i 6 = j where the product is over all i 2 { 1 , . . . , 2014 } except i = j . Evaluate 1 1 1
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- · · · + . b b b 1 2 2014
解析
- [ 17 ] For 1 ≤ j ≤ 2014, define 2014 ∏ 2014 2014 2014 b = j ( i − j ) j i =1 ,i 6 = j where the product is over all i ∈ { 1 , . . . , 2014 } except i = j . Evaluate 1 1 1
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- · · · + . b b b 1 2 2014 1 Answer: We perform Lagrange interpolation on the polynomial P ( x ) = 1 through the 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 j 1 = P (0) = , ∏ 2014 2013 2014 2014 ( − 1) ( i − j ) i =1 ,i 6 = j j =1 which equals ( ) 2014 ∑ 1 1 1 1 2014 2014 2014! = 2014! + + · · · + , ∏ 2014 2014 2014 2014 b b b j ( i − j ) 1 2 2014 i =1 ,i 6 = j j =1 1 so the desired sum is . 2014 2014!