Unformatted text preview: k for which k n + k 21 divides k m + k1. Problem 4 . The positive divisors of the integer n > 1 are d 1 < d 2 < ... < d k , so that d 1 = 1 ,d k = n . Let d = d 1 d 2 + d 2 d 3 + ··· + d k1 d k . Show that d < n 2 and ﬁnd all n for which d divides n 2 . Problem 5 . Find all realvalued functions on the reals such that ( f ( x ) + f ( y ))(( f ( u ) + f ( v )) = f ( xuyv ) + f ( xv + yu ) for all x,y,u,v . Problem 6 . n > 2 circles of radius 1 are drawn in the plane so that no line meets more than two of the circles. Their centers are O 1 ,O 2 , ··· ,O n . Show that ∑ i<j 1 /O i O j ≤ ( n1) π/ 4. 1...
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This note was uploaded on 09/07/2011 for the course RESEARCH R 101 taught by Professor T.s. during the Fall '11 term at Research College.
 Fall '11
 T.S.

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