[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/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/1

IGO 2016 Medium/1

For discussing Olympiad level Geometry Problems
Facebook Twitter

IGO 2016 Medium/1

Post Number:#1  Unread postby Thamim Zahin » Tue Jan 10, 2017 3:52 pm

1. In trapezoid $ABCD$ with $AB \parallel CD$, $w_1$ and $w_2$ are two circles with diameters $AD$ and $BC$, respectively. Let $X$ and $Y$ be two arbitrary points on $w_1$ and $w_2$, respectively. Show that the length of segment $XY$ is not more than half of the perimeter of $ABCD$.
I think we judge talent wrong. What do we see as talent? I think I have made the same mistake myself. We judge talent by the trophies on their showcases, the flamboyance the supremacy. We don't see things like determination, courage, discipline, temperament.
User avatar
Thamim Zahin
 
Posts: 97
Joined: Wed Aug 03, 2016 5:42 pm

Re: IGO 2016 Medium/1

Post Number:#2  Unread postby dshasan » Sat Jan 14, 2017 12:37 am

$Claim 1:$ In a trapezoid $ABCD (AB \parallel CD), AB + CD = 2EF $ where $E,F$ are the midpoints of $AD, BC$
$Proof:$ Extend $AD, BC$ so that they meet at $X$. Now, let $XA = a, XB = b, AE = ED = c, BF = FC = d$.
Now, $\frac{AB}{EF} = \frac{a}{a+c}$ and $\frac{EF}{CD} = \frac{a+c}{a+2c}$. From this two equation, we get, \[ AB + CD = 2EF\]
Now, segment $XY$ is largest when $XY$ passes through the midpoints of $AD$ and $BC$. Let $m,n$ be the midpoints of $AD$ and $BC$ respectively. SO, $Xm = \frac{1}{2} AD$ and $Yn = \frac{1}{2} BC$.
Therefore, $XY = Xm + mn + Yn = \frac{1}{2} ( AB + BC + CD + DA)$. For all other $XY$, it is less than half of the perimeter of $ABCD$
The study of mathematics, like the Nile, begins in minuteness but ends in magnificence.

- Charles Caleb Colton
dshasan
 
Posts: 60
Joined: Fri Aug 14, 2015 6:32 pm
Location: Dhaka,Bangladesh


Share with your friends: Facebook Twitter

  • Similar topics
    Replies
    Views
    Author

Return to Geometry

Who is online

Users browsing this forum: Google [Bot] and 2 guests

cron