IMO Shortlist 2009 problem N2
Kvaliteta:
Avg: 4,0Težina:
Avg: 6,0 A positive integer
is called balanced, if
or if
can be written as a product of an even number of not necessarily distinct primes. Given positive integers
and
, consider the polynomial
defined by
.
a) Prove that there exist distinct positive integers
and
such that all the number
,
, ...,
are balanced.
b) Prove that if
is balanced for all positive integers
, then
.
Proposed by Jorge Tipe, Peru
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
![N=1](/media/m/9/3/b/93bd8a08e3b33bb4b2cdbc003f3c4f83.png)
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
![P\!\left(x\right) = \left(x+a\right)\left(x+b\right)](/media/m/2/a/7/2a7943956f59564f5f2f81194915eb05.png)
a) Prove that there exist distinct positive integers
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
![P\!\left(1\right)](/media/m/3/7/e/37e3e75476b7f8197813eb828aaa6068.png)
![P\!\left(2\right)](/media/m/4/c/9/4c9e72727d41dc4367be4bf785664277.png)
![P\!\left(50\right)](/media/m/c/8/1/c81300af6fafbe625becbce7d4af8a5b.png)
b) Prove that if
![P\!\left(n\right)](/media/m/4/8/0/4801d7724878957965c43a39a261a5dd.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![a=b](/media/m/a/9/2/a92b57ffecf4f08b70899188d461ba5f.png)
Proposed by Jorge Tipe, Peru
Izvor: Međunarodna matematička olimpijada, shortlist 2009