HMMT 二月 2009 · 代数 · 第 7 题

英文解析

7. [ 5 ] Simplify the product
   100 100
   n + m n + m +2 2 n +1 2 m +1
   ∏ ∏
   x + x + x + x
   .
   2 n n + m 2 mx + 2 x + xm =1 n =1
   Express your answer in terms of x .
   ( )
   1002
   1+ x 1 1 1
   9900 9900 10000 10100
   Answer: x (OR x + x + x )
   2 4 2 4
   Solution: We notice that the numerator and denominator of each term factors, so the product isequal to
   100 100
   m n +1 m +1 n
   ∏ ∏
   ( x + x )( x + x )
   .
   m n 2
   ( x + x )
   m =1 n =1
   Each term of the numerator cancels with a term of the denominator except for those of the formm 101 101 n
   ( x + x ) and ( x + x ) for m, n = 1 , . . . , 100, and the terms in the denominator which remain are
   1 n 1 mof the form ( x + x ) and ( x + x ) for m, n = 1 , . . . , 100. Thus the product simplifies to
   ( )
   1002
   m 101
   ∏
   x + x
   1 mx + xm =1
   Reversing the order of the factors of the numerator, we find this is equal to
   ( ) ( )
   2 2
   100 100
   101 − m 101 1 m +1
   ∏ ∏
   x + x x + x
   100 − m = x
   1 m 1 mx + x x + xm =1 m =1
   ( )
   1002
   1 1
   ∏
   x + x 01
   100 − m = x
   1 1
   x + xm =1
   ( )
   1002
   99 · 100 1 + x = ( x )22
   as desired.2
   2 2 3 3