HMMT 二月 2010 · TEAM1 赛 · 第 5 题

英文解析

5. [ 20 ] Show that, for every positive integer n , there exists a monic polynomial of degree n with integercoefficients such that the coefficients are decreasing and the roots of the polynomial are all integers.
   Solution: We claim we can find values a and b such that p ( x ) = ( x − a )( x + b ) is a polynomial ofndegree n + 1 that satisfies these constraints. We show that its coefficients are decreasing by finding ageneral formula for the coefficient of x .k
   Team Round A
   ( ) ( )
   n nk k k − 1 n
   The coefficient of x is b − ab , which can be seen by expanding out ( x + b ) and thenk k − 1
   multiplying by ( x − a ). Then we must prove that
   ( ) ( ) ( ) ( )
   n n n nk +1 k k k − 1
   b − ab < b − ab ,
   k + 1 k k k − 1
   ( ( ) ( )) ( ( ) ( ))orn n n nk − 1 kab b − > b b − .
   k k − 1 k + 1 k
   ( )
   ( )nk
   Choose b > max in order to make sure the right-hand term in each product on each side ofn
   ( )
   k − 1
   the inequality sign is positive (we’ll be dividing by it, so this makes things much easier), and choose
   ( )
   n nb b −
   ( ( ) ( ))
   k +1 ka > max to make sure the inequality always holds. Since there are only finite valuesn nb −
   ( ) ( )
   k k − 1
   that k can take on given a fixed n (namely, integers between 0 and n inclusive), we can always findvalues of a and b that satisfy these constraints.