quiz7 - integers m 1 ,m 2 ,m 3 ,m 4 ,m 5 such that the...

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Putnam Exam Seminar Quiz 7 Fall 2010 November 1 Problem 1. Suppose p ( x ) is a polynomial of degree seven such that ( x - 1) 4 is a factor of p ( x ) + 1 and ( x + 1) 4 is a factor of p ( x ) - 1. Find p ( x ). Problem 2. Let k be a fixed positive integer. The n -th derivative of 1 x k - 1 has the form P n ( x ) ( x k - 1) n +1 where P n ( x ) is a polynomial. Find P n (1). [Putnam Exam, 2002, A1] Problem 3. Consider all lines which meet the graph of y = 2 x 4 + 7 x 3 + 3 x - 5 in four different points, sat ( x i ,y i ) for i = 1 , 2 , 3 , 4. Show that x 1 + x 2 + x 3 + x 4 4 is indepen- dent of the line and find its value. Problem 4. Let k be the smallest positive integer with the property that there are distinct
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Unformatted text preview: integers m 1 ,m 2 ,m 3 ,m 4 ,m 5 such that the polynomial p ( x ) = ( x-m 1 )( x-m 2 )( x-m 3 )( x-m 4 )( x-m 5 ) has exactly k nonzero coecients. Find, with proof, a set of integers m 1 ,m 2 ,m 3 ,m 4 ,m 5 for which this minimum k is achieved. [Putnam Exam, 1985, B1] Problem 5. Find a nonzero polynomial P ( x,y ) such that P ( b a c , b 2 a c ) = 0 for all real numbers a . ( Note: b c is the greatest integer less than or equal to .) [Putnam Exam, 2005, B1]...
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