[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 - IGO 2016 Medium/3
Page 1 of 1

IGO 2016 Medium/3

Unread postPosted: Tue Jan 10, 2017 3:59 pm
by Thamim Zahin
3. Find all positive integers $N$ such that there exists a triangle which can be dissected into $N$ similar quadrilaterals.

Re: IGO 2016 Mediam/3

Unread postPosted: Wed Jan 11, 2017 4:06 pm
by Thamim Zahin
It is possible for all integer $N \ge 3$

It is obvious that a triangle can't be partitioned in $1$ quadrilateral.

Also if we divide a triangle into $2$ quadrilaterals, one is convex but other one is not.

For $N=3$, take a equilateral triangle. And divide the triangle into $3$ congruent quadrilaterals. Like the figure. $O$ is the circumcenter. We draw $\angle OFB$ such that it is equal to $60^o$. Also same thing for $ \angle ODC $ and $ \angle OEA$.

It is easy to prove that the quadrilaterals are congruent.

Now we can make a quadrilateral $BCYX$ such that it is similar to $EOFB$. And $\angle ABX = \angle ACY = 60^o+120^o=180^o$. So that means $\triangle AXY$ is a new triangle with $5$ similar quadrilaterals. We can make this process over and over. So it is possible for all $N \ge 3$