[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 122: include(/home/shoeb/public_html/www.matholympiad.org.bd/forum/includes/phpbb-latex.php) [function.include]: failed to open stream: No such file or directory
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 122: include() [function.include]: Failed opening '/home/shoeb/public_html/www.matholympiad.org.bd/forum/includes/phpbb-latex.php' for inclusion (include_path='.:/opt/php53/lib/php')
[phpBB Debug] PHP Warning: in file [ROOT]/includes/session.php on line 1042: Cannot modify header information - headers already sent by (output started at [ROOT]/includes/functions.php:3887)
[phpBB Debug] PHP Warning: in file [ROOT]/includes/functions.php on line 4786: Cannot modify header information - headers already sent by (output started at [ROOT]/includes/functions.php:3887)
[phpBB Debug] PHP Warning: in file [ROOT]/includes/functions.php on line 4788: Cannot modify header information - headers already sent by (output started at [ROOT]/includes/functions.php:3887)
[phpBB Debug] PHP Warning: in file [ROOT]/includes/functions.php on line 4789: Cannot modify header information - headers already sent by (output started at [ROOT]/includes/functions.php:3887)
[phpBB Debug] PHP Warning: in file [ROOT]/includes/functions.php on line 4790: Cannot modify header information - headers already sent by (output started at [ROOT]/includes/functions.php:3887)
BdMO Online Forum • View topic - IMO $2017$ P$1$

IMO $2017$ P$1$

Discussion on International Mathematical Olympiad (IMO)
Facebook Twitter

IMO $2017$ P$1$

Post Number:#1  Unread postby ahmedittihad » Wed Jul 19, 2017 12:25 am

For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as
$$a_{n+1} =
\begin{cases}
\sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\
a_n + 3 & \text{otherwise.}
\end{cases}
$$
Determine all values of $a_0$ so that there exists a number $A$ such that $a_n = A$ for infinitely many values of $n$.
User avatar
ahmedittihad
 
Posts: 147
Joined: Mon Mar 28, 2016 6:21 pm

Re: IMO $2017$ P$1$

Post Number:#2  Unread postby Katy729 » Fri Aug 04, 2017 1:25 pm

Anyone?? :(
Katy729
 
Posts: 37
Joined: Sat May 06, 2017 2:30 am

Re: IMO $2017$ P$1$

Post Number:#3  Unread postby Katy729 » Fri Sep 15, 2017 3:11 am

Please someone post a solution... :( :(
Katy729
 
Posts: 37
Joined: Sat May 06, 2017 2:30 am

Re: IMO $2017$ P$1$

Post Number:#4  Unread postby ahmedittihad » Sun Oct 01, 2017 10:25 pm

Case 1: $a_0\equiv 0\pmod{3}$.
We have $a_m\equiv 0\pmod{3}\,\,\forall m\geq 0$.
If $a_0=3$ then $a_{3m}=3\,\,\forall m\geq 0$, therefore $a_0=3$ satisfying the condition of the problem.
If $a_0=3k$ for some $k>1$. We will prove that there is an index $m_0$ such that $a_{m_0}<a_0$, and therefore (by replace $a_0$ by $a_{m_0}$) $a_0$ satisfying the condition of the problem. If $a_0$ is a square then $a_1<a_0$ else $m^2<3k<(m+1)^2$ for some positive integer $m$, exactly one of $m+1$, $m+2$, $m+3$ is divisible $3$, assume that it is $m+x\,\, (x\in\{0,1,2\})$, for some $m_0$ we have $a_{m_0}=m+x\leq m+3<3k=a_0$, we're done.

Case 2: $a_0\equiv 2\pmod{3}$.
In this case we have $a_n=a_0+3n\,\,\forall n\geq 0$ therefore $a_0$ does not meet the condition of the problem.

Case 3: $a_0\equiv 1\pmod{3}$.
We have $a_m\not\equiv 0\pmod{3}\,\,\forall m\geq 0$.
If $a_m\equiv 2\pmod{3}$ for some $m$ then by case 2, the sequence is unbound, therefore $a_0$ does not meet the condition of the problem.
If $a_m\equiv 1\pmod{3}\,\,\forall m\geq 0$ then assume that $a_k$ is smallest in $\{a_n\}$. Similary case 1, we can find $a_l$ such that $a_l<a_k$, contradiction! Therefore $a_0$ does not meet the condition of the problem.

Conclude, $a_0\equiv 0\pmod{3}$.[/
Frankly, my dear, I don't give a damn.
User avatar
ahmedittihad
 
Posts: 147
Joined: Mon Mar 28, 2016 6:21 pm

Re: IMO $2017$ P$1$

Post Number:#5  Unread postby Katy729 » Mon Oct 02, 2017 3:00 pm

Beautifull solution ahmedittihad!! :)
Katy729
 
Posts: 37
Joined: Sat May 06, 2017 2:30 am


Share with your friends: Facebook Twitter

  • Similar topics
    Replies
    Views
    Author

Return to International Mathematical Olympiad (IMO)

Who is online

Users browsing this forum: Baidu [Spider] and 2 guests

cron