HMMT 十一月 2011 · GEN 赛 · 第 10 题

英文解析

10. [ 8 ] Let r , r , . . . , r be the distinct complex roots of the polynomial P ( x ) = x − 7. Let
    1 2 7
    ∏
    K = ( r + r ),
    i j
    1 ≤ i<j ≤ 7
    that is, the product of all numbers of the form r + r , where i and j are integers for which 1 ≤ i < j ≤ 7.
    i j
    Determine the value of K .2
    Answer: 117649 We first note that x − 7 = ( x − r )( x − r ) · · · ( x − r ), which implies, replacing 7
    1 2 7
    x by − x and taking the negative of the equation, that ( x + r )( x + r ) · · · ( x + r ) = x + 7. Also note 7
    1 2 7
    that the product of the r is just the constant term, so r r · · · r = 7.
    i 1 2 7
    General Test
    Now, we have that
    ∏7
    7 2 2
    2 · 7 · K = ( 2 r ) Kii =1
    ∏ ∏7 = 2 r ( r + r )2
    i i ji =1
    1 ≤ i<j ≤ 7
    ∏ ∏ ∏
    = ( r + r ) ( r + r ) ( r + r )
    i j i j i j
    1 ≤ i = j ≤ 7 1 ≤ i<j ≤ 7 1 ≤ j<i ≤ 7
    ∏
    = ( r + r )
    i j
    1 ≤ i,j ≤ 7
    7 7
    ∏ ∏
    = ( r + r ) .
    i ji =1 j =1
    ∏7
    However, note that for any fixed i , ( r + r ) is just the result of substuting x = r into ( x + r )( x +
    i j i 1
    j =1
    r ) · · · ( x + r ). Hence,
    2 7
    ∏7
    7 7
    ( r + r ) = r + 7 = ( r − 7) + 14 = 14 .
    i ji ij =1
    7 2 6
    Therefore, taking the product over all i gives 14 , which yields K = 7 = 117649.
    General Test