Problem 1. 
     Denote by   w(m)   the minimal positive 
  integer   n    for which   m | f(n)   where  f(n)   stands
  for the n-th term of Fibonacci sequence  
    {1,1,2,3,5,8,13,21,34,55,89,...}.

   For example,   w(1)=1, w2)=3,  w(3)=4, w(4)=6,  w(5)=5.  
   Prove that    w(m) < 3m,  for all   m.
   May the constant   3   be improved  (replaced by a 
       smaller one)?      (10 pts.)
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Problem 2. 
     Assume that   p(x)= x^5+ ...  is a monic polynomial of 
degree  5   over integers.  Prove that for every  integer   
m >120=5!   there exists  an integer   n   such that    p(n)   
is not a multiple of  m.       (2 pts.)
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Problem 3.
   For  positive integers   a,b,   we denote by   S(a,b)
 the set  of all possible sums  of the numbers  a  and  b,  
 with arbitrary multiplicities.  For example,

  S(4,7)={4,7,8,11,12,14,15,16,18,19,20,21,22,23,24,...}.

Assume that    a,b  are two positive integers such  
   that   (a,b)=1.  Let   c = ab-a-b.

Problem 3A.  Prove that   c   is not in the set   S(a,b).
Problem 3B.  Prove that  every  integer  larger than   c   
   belongs  to  S(a,b).         (3 pts.)
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