IMO Shortlist 2002 problem C1
Kvaliteta:
Avg: 4,0Težina:
Avg: 6,0 Let
be a positive integer. Each point
in the plane, where
and
are non-negative integers with
, is coloured red or blue, subject to the following condition: if a point
is red, then so are all points
with
and
. Let
be the number of ways to choose
blue points with distinct
-coordinates, and let
be the number of ways to choose
blue points with distinct
-coordinates. Prove that
.
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![(x,y)](/media/m/c/9/1/c91aec4078b932368ded863349deaec5.png)
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
![y](/media/m/c/c/0/cc082a07a517ebbe9b72fd580832a939.png)
![x+y<n](/media/m/4/7/0/4701862aae6ca17faedc3b95ce917048.png)
![(x,y)](/media/m/c/9/1/c91aec4078b932368ded863349deaec5.png)
![(x',y')](/media/m/1/5/8/15890085966de63382dce378911030d1.png)
![x'\leq x](/media/m/f/f/d/ffd4a6c512bf761b83607175e703fa5e.png)
![y'\leq y](/media/m/8/7/e/87e10f8a17aeb81df3abde81bd60a8fa.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![y](/media/m/c/c/0/cc082a07a517ebbe9b72fd580832a939.png)
![A=B](/media/m/6/8/9/689f49b3b57cfd7b358bce6b57563e11.png)
Izvor: Međunarodna matematička olimpijada, shortlist 2002