IMO Shortlist 1994 problem G4
Dodao/la:
arhiva2. travnja 2012. Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be an isosceles triangle with
![AB = AC](/media/m/9/d/3/9d3b4c44cee3883e9b08931f9d15c670.png)
.
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
is the midpoint of
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
and
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
is the point on the line
![AM](/media/m/9/2/1/921d54bb92ada2d2120b2591b722ea12.png)
such that
![OB](/media/m/5/0/3/503e9123196089d1244989e870075ca4.png)
is perpendicular to
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
.
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
is an arbitrary point on
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
different from
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
and
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
.
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
lies on the line
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
and
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
lies on the line
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
such that
![E, Q, F](/media/m/3/4/6/3468bc6228f46c502f5691dfa7f881bf.png)
are distinct and collinear. Prove that
![OQ](/media/m/6/e/2/6e2bb12104a86753c8bd6e1dfc012cc1.png)
is perpendicular to
![EF](/media/m/f/5/5/f5594d5ec47ea777267cf010e788fedd.png)
if and only if
![QE = QF](/media/m/2/3/f/23f885b7cbb707032990e520666974d2.png)
.
%V0
Let $ABC$ be an isosceles triangle with $AB = AC$. $M$ is the midpoint of $BC$ and $O$ is the point on the line $AM$ such that $OB$ is perpendicular to $AB$. $Q$ is an arbitrary point on $BC$ different from $B$ and $C$. $E$ lies on the line $AB$ and $F$ lies on the line $AC$ such that $E, Q, F$ are distinct and collinear. Prove that $OQ$ is perpendicular to $EF$ if and only if $QE = QF$.
Izvor: Međunarodna matematička olimpijada, shortlist 1994