IMO Shortlist 1994 problem G2
Dodao/la:
arhiva2. travnja 2012. ![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
is a quadrilateral with
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
parallel to
![AD](/media/m/6/9/6/69672822808d046d0e94ab2fa7f2dc80.png)
.
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
is the midpoint of
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
,
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
is the midpoint of
![MA](/media/m/1/a/2/1a298617d983de609dc306755b66a265.png)
and
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
is the midpoint of
![MB](/media/m/5/d/c/5dcd44e421f0875ab7ce126cece0fe27.png)
. The lines
![DP](/media/m/f/d/8/fd8300e37d26cfa47fa724a9df058301.png)
and
![CQ](/media/m/a/8/4/a846b659c7be23bc98305e03c8f65850.png)
meet at
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
. Prove that
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
is inside the quadrilateral
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
.
%V0
$ABCD$ is a quadrilateral with $BC$ parallel to $AD$. $M$ is the midpoint of $CD$, $P$ is the midpoint of $MA$ and $Q$ is the midpoint of $MB$. The lines $DP$ and $CQ$ meet at $N$. Prove that $N$ is inside the quadrilateral $ABCD$.
Izvor: Međunarodna matematička olimpijada, shortlist 1994