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# quiz11 - x 3 3 xy y 3 = 1 contains only one set of three...

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Putnam Exam Seminar Quiz 11 Fall 2010 December 1 Problem 1. A game involves jumping to the right on the real number line. If a and b are real numbers and b > a , the cost of jumping from a to b is b 3 - ab 2 . For what real numbers c can one travel from 0 to 1 in a finite number of jumps with total cost exactly c ? [Putnam Exam, 2009, B2] Problem 2. Let f : R 2 R be a function such that f ( x, y ) + f ( y, z ) + f ( z, x ) = 0 for all real numbers x , y and z . Prove that there exists a function g : R R such that f ( x, y ) = g ( x ) - g ( y ) for all real numbers x and y . [Putnam Exam, 2008, A1] Problem 3. Let f be a nonconstant polynomial with positive integer coefficients. Prove that if n is a positive integer, then f ( n ) divides f ( f ( n ) + 1) if and only if n = 1. [Putnam Exam, 2007, B1] Problem 4. Show that the curve
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Unformatted text preview: x 3 + 3 xy + y 3 = 1 contains only one set of three distinct points A , B and C which are the vertices of an equilateral triangle, and ﬁnd its area. [Putnam Exam, 2006, B1] Problem 5. Deﬁne a function f on the real numbers by f ( x ) = ( if x is irrational, 1 /q if x = p/q with p ∈ Z ,q ∈ N , gcd( p,q ) = 1 . Determine the set of points on which f is continuous. Problem 6. Deﬁne a sequence { u n } ∞ n =0 by u = u 1 = u 2 = 1, and thereafter by the condition that det ± u n u n +1 u n +2 u n +3 ² = n ! for all n ≥ 0. Show that u n is an integer for all n . (By convention, 0! = 1.) [Putnam Exam, 2004, A3]...
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