7ᵗʰ Iranian Geometry Olympiad 2020 (Intermediate) P1

For discussing Olympiad level Geometry Problems
IftakharTausifFarhan
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7ᵗʰ Iranian Geometry Olympiad 2020 (Intermediate) P1

Unread post by IftakharTausifFarhan » Sat Dec 12, 2020 4:07 pm

A trapezoid $ABCD$ is given where $AB$ and $CD$ are parallel. Let $M$ be the midpoint of the segment $AB$. Point $N$ is located on the segment $CD$ such that $\angle ADN= \frac{1}{2} \angle MNC$ and $\angle BCN=\frac{1}{2} \angle MND$. Prove that $N$ is the midpoint of segment $CD$.

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Mehrab4226
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Re: 7ᵗʰ Iranian Geometry Olympiad 2020 (Intermediate) P1

Unread post by Mehrab4226 » Fri Dec 18, 2020 10:32 pm

It took 1 hour at the exam hall and 25 minutes to recall it about a decade later :)
Let us draw the bisectors of $\angle MND $ and $\angle MNC$ which intersects AB at X and Y respectively.
we get the followings:
$\angle ADN=\angle YNC= \angle MNY =\angle MYN$[Using AB||CD(YN transversal) and what is given]
$\angle BCN=\angle XND = \angle MNX = \angle MXN$[Same reason as before]
Now, $\triangle MNX$ and $\triangle MNY$ are isosceles triangles.

$\therefore$ $XM= MN$ and $MY=YN$
$\therefore XM= YN$
And we know, $AM= MB$
$\therefore AY=BX \cdots (1)$


Again ,
$\angle ADN=\angle YNC$ $\therefore AD||NY$ and $AY||DN(Given)$
$\therefore$ $ADNY$ is a parallelogram.
$\therefore AY=DN $
Similarly, $BCNX$ is a parallelogram.
$\therefore BX=NC$
From (1) we get $AY=BX$
$\therefore DN=CN$
$\therefore$ N is the midpoint of CD[proved]
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Asif Hossain
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Re: 7ᵗʰ Iranian Geometry Olympiad 2020 (Intermediate) P1

Unread post by Asif Hossain » Tue Feb 23, 2021 10:11 pm

Just an off topic question: @ mehrab did you participate in IGO 2020? how did you participate??
Hmm..Hammer...Treat everything as nail

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Mehrab4226
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Re: 7ᵗʰ Iranian Geometry Olympiad 2020 (Intermediate) P1

Unread post by Mehrab4226 » Tue Feb 23, 2021 11:39 pm

Asif Hossain wrote:
Tue Feb 23, 2021 10:11 pm
Just an off topic question: @ mehrab did you participate in IGO 2020? how did you participate??
yes, I did. There was a google docs form for registration posted on the BDMO FB page.
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

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