IMO Shortlist 2010 problem N4
Kvaliteta:
Avg: 0,0Težina:
Avg: 7,0 Let
be integers, and let
For any positive integer
we say that the pair
is
-good if
implies
for all integers
We say that
is
if
is
-good for infinitely many positive integers ![n.](/media/m/4/7/8/478d25bfe04800537ae6e85be9d11ea2.png)
(a) Find a pair
which is 51-good, but not very good.(b) Show that all 2010-good pairs are very good.
Proposed by Okan Tekman, Turkey
![a, b](/media/m/a/2/b/a2bdbf048e2daac0a021a1d79f6fb9bf.png)
![P(x) = ax^3+bx.](/media/m/5/d/e/5def6df57bee77b029d13850af2d848b.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![(a,b)](/media/m/e/2/6/e263229694cdbeb908488db2d0351f0a.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![n | P(m)-P(k)](/media/m/c/9/6/c9657d501e21c72bb309ce5015ebd961.png)
![n | m - k](/media/m/0/5/2/052a90cb2b0ff63e44c6e01e6e51ab36.png)
![m, k.](/media/m/d/b/3/db31c4d06f2eab8f3e2728fd40307144.png)
![(a,b)](/media/m/e/2/6/e263229694cdbeb908488db2d0351f0a.png)
![very \ good](/media/m/c/6/5/c652247356514a948b4c4f767f1dc1ac.png)
![(a,b)](/media/m/e/2/6/e263229694cdbeb908488db2d0351f0a.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![n.](/media/m/4/7/8/478d25bfe04800537ae6e85be9d11ea2.png)
(a) Find a pair
![(a,b)](/media/m/e/2/6/e263229694cdbeb908488db2d0351f0a.png)
Proposed by Okan Tekman, Turkey
Izvor: Međunarodna matematička olimpijada, shortlist 2010