# 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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 views: 10 posted: 6/3/2012 language: pages: 1
Lingjuan Ma