hw9 - Putnam Exam Seminar Fall 2010 Assignment 9 Due...

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Putnam Exam Seminar Assignment 9 Fall 2010 Due November 8 Exercise 1. Do there exist polynomials a ( x ) ,b ( x ) ,c ( y ) ,d ( y ) such that 1 + xy + x 2 y 2 = a ( x ) c ( y ) + b ( x ) d ( y ) holds identically? [Putnam Exam, 2003, B1] Exercise 2. Let n be a positive integer and define f ( n ) = 1! + 2! + 3! + ··· + n ! . Find polynomials P ( x ) and Q ( x ) such that f ( n + 2) = P ( n ) f ( n + 1) + Q ( n ) f ( n ) for all n 1. Exercise 3. Let P ( x ) = c n x n + c n - 1 x n - 1 + ··· + c 0 be a polynomial with integer coefficients. Suppose that r is a rational number such that P ( r ) = 0. Show that the n numbers c n r,c n r 2 + c n - 1 r,c n r 3 + c n - 1 r 2 + c n - 1 r,. ..,c n r n + c n - 1 r n - 1 + ··· c 1 r are integers. [Putnam Exam, 2004, B1] Exercise 4.
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This note was uploaded on 02/29/2012 for the course MATH 1190 taught by Professor Ryandaileda during the Fall '10 term at Trinity University.

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