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Unformatted text preview: + 1 ,n + 2 ,...,n + 6 } can be divided into two sets with the product of elements in one set equal to the product of elements in the other set. 5. Put 1983. A3. Let p be an odd prime, and let F ( n ) = 1 + 2 n + 3 n 2 + ... + ( p1) n p2 . Prove that if a and b are not congruent modulo P then F ( a ) and F ( b ) are not congruent modulo p . 6. IMO 2002. The positive divisors of an integer n > 1 are 1 = d 1 < d 2 < ... < d k = n . Let s = d 1 d 2 + d 2 d 3 + ... + d k1 d k . Prove that s < n 2 and ﬁnd all n for which s divides n 2 . 7. Put 2001. A5. Prove that there are unique positive integers a,n such that a n +1( a +1) n = 2001 . 8. Put 1996. A6. The sequence a n is deﬁned by a 1 = 1 ,a 2 = 2 ,a 3 = 24 , and, for n ≥ 4 , a n = 6 a 2 n1 a n38 a n1 a 2 n2 a n2 a n3 Show that, for all n , a n is an integer multiple of n ....
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 Fall '14
 NORIN
 Math, Number Theory, Integers, Prime number, positive integers, congruences, congruent modulo

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