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# hw7 - = x f x for all x> 0[Putnam Exam 2005 B-3 Exercise...

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Putnam Exam Seminar Assignment 7 Fall 2010 Due October 25 Exercise 1. Functions f , g , h are differentiable on some open interval around 0 and satisfy the equations and initial conditions f 0 = 2 f 2 gh + 1 gh , f (0) = 1 , g 0 = fg 2 h + 4 fh , g (0) = 1 , h 0 = 3 fgh 2 + 1 fg , h (0) = 1 . Find an explicit formula for f ( x ), valid in some open interval around 0. [Putnam Exam, 2009, A-2] Exercise 2. Define polynomials f n ( x ) for n 0 by f 0 ( x ) = 1, f n (0) = 0 for n 1, and d dx f n +1 ( x ) = ( n + 1) f n ( x + 1) for n 0. Find, with proof, the explicit factorization of f 100 (1) into powers of distinct primes. [Putnam Exam, 1985, B-2] Exercise 3. Find all differentiable functions f : (0 , ) (0 , ) for which there is a positive real number a such that f 0 a x =
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Unformatted text preview: = x f ( x ) for all x > 0. [Putnam Exam, 2005, B-3] Exercise 4. Let f be a real function on the real line with a continuous third derivative. Prove that there exists a point a such that f ( a ) · f ( a ) · f 00 ( a ) · f 000 ( a ) ≥ 0. [Putnam Exam, 1998, A-3] Exercise 5. Let p ( x ) be a nonzero polynomial of degree less than 1992 having no noncon-stant factor in common with x 3-x . Let d 1992 dx 1992 ³ p ( x ) x 3-x ´ = f ( x ) g ( x ) for polynomials f ( x ) and g ( x ). Find the smallest possible degree of f ( x ). [Putnam Exam, 1992, B-4]...
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