IMO Shortlist 2004 problem C5
Dodao/la:
arhiva2. travnja 2012. 3.
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
and
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
play a game, given an integer
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
,
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
writes down
![1](/media/m/a/9/1/a913f49384c0227c8ea296a725bfc987.png)
first, then every player sees the last number written and if it is
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
then in his turn he writes
![n+1](/media/m/2/a/7/2a7327e09a84d01a602088c9f045cbde.png)
or
![2n](/media/m/d/2/d/d2da874dc9bc356be9468cdbd57fbfdf.png)
, but his number cannot be bigger than
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
. The player who writes
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
wins!, For wich values of
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
does
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
wins?
%V0
3. $A$ and $B$ play a game, given an integer $N$, $A$ writes down $1$ first, then every player sees the last number written and if it is $n$ then in his turn he writes $n+1$ or $2n$, but his number cannot be bigger than $N$. The player who writes $N$ wins!, For wich values of $N$ does $B$ wins?
Izvor: Međunarodna matematička olimpijada, shortlist 2004