International Mathematical Olympiad 2000

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					                International Mathematical Olympiad 2000
                        Hong Kong Team Training
                                  Test 2

Time allowed: 3 hours.


1.   The sequence {an }1 is defined by a1  1 , a2  2 , a3  24 , and for n  4 ,
                       n


                                           6an1an3  8an1an2
                                              2              2
                                      an                        .
                                                 an2 an3
     Show that for all n, a n     is an integer multiple of n.



2.   Let p( x) be a polynomial with integer coefficients and degree higher than 4.
     Suppose the roots of p( x) are distinct integers, and 0 is a root of p( x) . Find
     all integer roots of p( p( x)) in terms of the roots of p( x) .
     Can the assumption that deg p( x)  4 be relaxed?



3.   Suppose x1 , x2 ,   , xm , y1 , y2 ,    , yn are integers satisfying the following
     inequalities
     (a) 1  x1  x2   xm  y1  y2   yn
     (b) x1  x2   xm  y1  y2   yn
     Prove that x1 x2 xm  y1 y2 yn .
     Can the assumption that “ 1  x1 ” be relaxed to “ 1  x1 ”?



4.   Let A be a finite set of integers, each greater than 1. Suppose that for each integer
     n, there is some s  A such that ( s, n)  1 or ( s, n)  s . Show that there exist
     u, v  A such that (u , v) is prime. [Recall that (a, b) denote the greatest
     common divisor of a and b.]




                                            ~ End of Test ~

				
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Lingjuan Ma Lingjuan Ma
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