HMMT 二月 2014 · 冲刺赛 · 第 28 题

英文解析

28. [ 17 ] Let f ( n ) and g ( n ) be polynomials of degree 2014 such that f ( n ) + ( − 1) g ( n ) = 2 for n =
    2014
    1 , 2 , . . . , 4030. Find the coefficient of x in g ( x ).
    2014
    Answer: Define the polynomial functions h and h by h ( x ) = f (2 x ) + g (2 x ) and 3
    2014 1 2 1
    2 · 2014!
    2 x 2 x − 1
    h ( x ) = f (2 x − 1) − g (2 x − 1). Then, the problem conditions tell us that h ( x ) = 2 and h ( x ) = 2
    2 1 2
    for x = 1 , 2 , . . . , 2015.
    By the Lagrange interpolation formula, the polynomial h is given by
    2015 20151
    ∑ ∏
    x − j
    2 ih ( x ) = 2 .
    i − j 1
    i =1 j =1
    i 6 = j
    2014
    So the coefficient of x in h ( x ) is
    ( )1
    2015 2015 2015
    2014
    ∑ ∏ ∑
    1 1 2014 4 · 3
    2 i 2 i 2015 − i
    2 = 2 ( − 1) =
    i − j 2014! i − 1 2014!
    i =1 j =1 i =1
    i 6 = jwhere the last equality follows from the binomial theorem. By a similar argument, the coefficient of
    2014
    2014 2 · 3
    x in h ( x ) is .
    2014!2
    1 2014
    We can write g ( x ) = ( h ( x/ 2) − h (( x + 1) / 2)). So, the coefficient of x in g ( x ) is
    1 2
    ( )2
    2014 2014 2014
    1 4 · 3 2 · 3 3 − = .
    2014 2014 2014
    2 2 · 2014! 2 · 2014! 2 · 2014!