IMO Shortlist 1994 problem N4
Dodao/la:
arhiva2. travnja 2012. Define the sequences
![a_n, b_n, c_n](/media/m/d/b/4/db4229eab4c0807771f4ecf3ca2f1418.png)
as follows.
![a_0 = k, b_0 = 4, c_0 = 1](/media/m/5/a/4/5a4b5e20864b5b72d3291d6801ebc9c2.png)
.
If
![a_n](/media/m/1/f/f/1ff6f81c68b9c6fb726845c9ce762d7a.png)
is even then
![a_{n + 1} = \frac {a_n}{2}](/media/m/4/d/e/4de25592aa5d1375516cf94ffac54756.png)
,
![b_{n + 1} = 2b_n](/media/m/6/f/8/6f8e4d200a12fbf3d4ae002ec77f3afa.png)
,
![c_{n + 1} = c_n](/media/m/5/6/0/56040e86c6133f0c297b64dcc35e46d0.png)
.
If
![a_n](/media/m/1/f/f/1ff6f81c68b9c6fb726845c9ce762d7a.png)
is odd, then
![a_{n + 1} = a_n - \frac {b_n}{2} - c_n](/media/m/a/f/f/affcae44603d704d4a5860ce1c6a348f.png)
,
![b_{n + 1} = b_n](/media/m/6/0/4/60423773c4c081ec3c94e192eca1feac.png)
,
![c_{n + 1} = b_n + c_n](/media/m/b/0/7/b07dfb4f16b2700c3c4341ce044684a3.png)
.
Find the number of positive integers
![k < 1995](/media/m/f/5/c/f5c9afaf694e05fb6e20ed015a57a245.png)
such that some
![a_n = 0](/media/m/d/7/9/d797ae5a8ea97f785bd6549071606cd3.png)
.
%V0
Define the sequences $a_n, b_n, c_n$ as follows. $a_0 = k, b_0 = 4, c_0 = 1$.
If $a_n$ is even then $a_{n + 1} = \frac {a_n}{2}$, $b_{n + 1} = 2b_n$, $c_{n + 1} = c_n$.
If $a_n$ is odd, then $a_{n + 1} = a_n - \frac {b_n}{2} - c_n$, $b_{n + 1} = b_n$, $c_{n + 1} = b_n + c_n$.
Find the number of positive integers $k < 1995$ such that some $a_n = 0$.
Izvor: Međunarodna matematička olimpijada, shortlist 1994