HMMT 二月 2009 · 代数 · 第 7 题

HMMT February 2009 — Algebra — Problem 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 m x + 2 x + x m =1 n =1 Express your answer in terms of x . 2 2 3 3

解析

[ 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 m x + 2 x + x m =1 n =1 Express your answer in terms of x . ( ) 2 100 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 is equal 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 form m 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 m of the form ( x + x ) and ( x + x ) for m, n = 1 , . . . , 100. Thus the product simplifies to ( ) 2 100 m 101 ∏ x + x 1 m x + x m =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 m x + x x + x m =1 m =1 ( ) 2 100 1 1 ∏ x + x 01 100 − m = x 1 1 x + x m =1 ( ) 2 100 99 · 100 1 + x 2 2 = ( x ) 2 as desired. 2 2 3 3