IMO Shortlist 1961 problem 5
Dodao/la:
arhiva2. travnja 2012. Construct a triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
if
![AC=b](/media/m/3/7/7/3774fbe4732f4438c90b6e0c85fdf715.png)
,
![AB=c](/media/m/1/7/3/173393364c0deb233785693ebc4e6156.png)
and
![\angle AMB=w](/media/m/b/f/1/bf1ec98aaec6d09b5112737b98d8ec11.png)
, where
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
is the midpoint of the segment
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
and
![w<90](/media/m/1/f/b/1fb6868a0e47a83b5f0969edf6305725.png)
. Prove that a solution exists if and only if
![b \tan{\dfrac{w}{2}} \leq c <b](/media/m/5/f/d/5fdb03511add55130c3ab4b899e291f0.png)
In what case does the equality hold?
%V0
Construct a triangle $ABC$ if $AC=b$, $AB=c$ and $\angle AMB=w$, where $M$ is the midpoint of the segment $BC$ and $w<90$. Prove that a solution exists if and only if $$b \tan{\dfrac{w}{2}} \leq c <b$$ In what case does the equality hold?
Izvor: Međunarodna matematička olimpijada, shortlist 1961