MEMO 2010 ekipno problem 1
Dodao/la:
arhiva28. travnja 2012. Three strictly increasing sequences
![a_1, a_2, a_3, \ldots,\qquad b_1, b_2, b_3, \ldots,\qquad c_1, c_2, c_3, \ldots](/media/m/1/6/5/165c6b85e1d90e74aa768f05dc66f596.png)
of positive integers are given. Every positive integer belongs to exactly one of the three sequences. For every positive integer
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
, the following conditions hold:
(a)
![c_{a_n}=b_n+1](/media/m/0/4/7/04734446ac685b01c975a9fb8d08fd56.png)
;
(b)
![a_{n+1}>b_n](/media/m/1/d/3/1d372a254805b1a4c1c1c3e7e505b21a.png)
;
(c) the number
![c_{n+1}c_{n}-(n+1)c_{n+1}-nc_n](/media/m/9/b/9/9b984b272542f85ed3e27e7461a31bd0.png)
is even.
Find
![a_{2010}](/media/m/a/6/b/a6b16c39ea78b386f9035928f3ca8e39.png)
,
![b_{2010}](/media/m/a/7/0/a701bc472c53177fa28930c9d5f1ee1a.png)
and
![c_{2010}](/media/m/7/a/a/7aa9c2d9e50d6cb7f8f41a7199746d39.png)
.
%V0
Three strictly increasing sequences
$$a_1, a_2, a_3, \ldots,\qquad b_1, b_2, b_3, \ldots,\qquad c_1, c_2, c_3, \ldots$$
of positive integers are given. Every positive integer belongs to exactly one of the three sequences. For every positive integer $n$, the following conditions hold:
(a) $c_{a_n}=b_n+1$;
(b) $a_{n+1}>b_n$;
(c) the number $c_{n+1}c_{n}-(n+1)c_{n+1}-nc_n$ is even.
Find $a_{2010}$, $b_{2010}$ and $c_{2010}$.
Izvor: Srednjoeuropska matematička olimpijada 2010, ekipno natjecanje, problem 1