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# quiz10 - c/d where a,b,c,d are integers[Putnam Exam 1989 B1...

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Putnam Exam Seminar Quiz 10 Fall 2010 November 29 Problem 1. Determine, with proof, the number of ordered triples ( A 1 , A 2 , A 3 ) of sets which have the property that (i) A 1 A 2 A 3 = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 } , and (ii) A 1 A 2 A 3 = . Express your answer in the form 2 a 3 b 5 c 7 d , where a, b, c, d are nonnegative integers. [Putnam Exam, 1985, A1] Problem 2. What is the units (i.e., rightmost) digit of 10 20000 10 100 + 3 ? [Putnam Exam, 1986, A2] Problem 3. If every point of the plane is painted one of three colors, do there necessarily exist two points of the same color exactly one inch apart? [Putnam Exam, 1988, A4(a)] Problem 4. A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to be hit, find the probability that the point hit is nearer to the center than to any edge. Express your answer in the form ( a

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Unformatted text preview: + c ) /d , where a,b,c,d are integers. [Putnam Exam, 1989, B1] Problem 5. Let T = 2 , T 1 = 3 , T 2 = 6 , and for n ≥ 3, T n = ( n + 4) T n-1-4 nT n-2 + (4 n-8) T n-3 . The ﬁrst few terms are 2 , 3 , 6 , 14 , 40 , 152 , 784 , 5168 , 40576 . Find, with proof, a formula for T n of the form T n = A n + B n , where { A n } and { B n } are well-known sequences. [Putnam Exam, 1990, A1] Problem 6. Suppose f and g are non-constant, diﬀerentiable, real-valued functions deﬁned on (-∞ , ∞ ). Furthermore, suppose that for each pair of real numbers x and y , f ( x + y ) = f ( x ) f ( y )-g ( x ) g ( y ) , g ( x + y ) = f ( x ) g ( y ) + g ( x ) f ( y ) . If f (0) = 0, prove that ( f ( x )) 2 + ( g ( x )) 2 = 1 for all x . [Putnam Exam, 1991, B2]...
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quiz10 - c/d where a,b,c,d are integers[Putnam Exam 1989 B1...

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