IMO Shortlist 1981 problem 15
Dodao/la:
arhiva2. travnja 2012. Consider a variable point
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
inside a given triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
. Let
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
,
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
,
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
be the feet of the perpendiculars from the point
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
to the lines
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
,
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
,
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
, respectively. Find all points
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
which minimize the sum
%V0
Consider a variable point $P$ inside a given triangle $ABC$. Let $D$, $E$, $F$ be the feet of the perpendiculars from the point $P$ to the lines $BC$, $CA$, $AB$, respectively. Find all points $P$ which minimize the sum $${BC\over PD}+{CA\over PE}+{AB\over PF}.$$
Izvor: Međunarodna matematička olimpijada, shortlist 1981