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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 n14 nT n2 + (4 n8) T n3 . 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 wellknown sequences. [Putnam Exam, 1990, A1] Problem 6. Suppose f and g are nonconstant, diﬀerentiable, realvalued 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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 Fall '10
 RyanDaileda
 Sets, Tn, Zagreb, Putnam Exam

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