IMO Shortlist 1969 problem 21
Dodao/la:
arhiva2. travnja 2012. ![(FRA 4)](/media/m/1/a/b/1abc6de76c4703bd2a3323c7221412b2.png)
A right-angled triangle
![OAB](/media/m/4/a/c/4ac8783af608ce16ab6fe8ecef768cd3.png)
has its right angle at the point
![B.](/media/m/6/b/c/6bc9b0e17086ccf6fe102db1f5c3ebcf.png)
An arbitrary circle with center on the line
![OB](/media/m/5/0/3/503e9123196089d1244989e870075ca4.png)
is tangent to the line
![OA.](/media/m/e/0/7/e07b5e02de0a3a96585f3cca91f2ecfb.png)
Let
![AT](/media/m/7/9/8/798e23717bf03c06947e5e3f7c9166ec.png)
be the tangent to the circle different from
![OA](/media/m/b/2/0/b206c115fb0e114a37cf644cba5338cb.png)
(
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
is the point of tangency). Prove that the median from
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
of the triangle
![OAB](/media/m/4/a/c/4ac8783af608ce16ab6fe8ecef768cd3.png)
intersects
![AT](/media/m/7/9/8/798e23717bf03c06947e5e3f7c9166ec.png)
at a point
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
such that
%V0
$(FRA 4)$ A right-angled triangle $OAB$ has its right angle at the point $B.$ An arbitrary circle with center on the line $OB$ is tangent to the line $OA.$ Let $AT$ be the tangent to the circle different from $OA$ ($T$ is the point of tangency). Prove that the median from $B$ of the triangle $OAB$ intersects $AT$ at a point $M$ such that $MB = MT.$
Izvor: Međunarodna matematička olimpijada, shortlist 1969