IMO Shortlist 1987 problem 5
Dodao/la:
arhiva2. travnja 2012. Find, with proof, the point
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
in the interior of an acute-angled triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
for which
![BL^2+CM^2+AN^2](/media/m/6/0/0/6000ffb70dc1111d37e0995f2f4bfcf3.png)
is a minimum, where
![L,M,N](/media/m/6/0/b/60bcd73f3b373c52cdb717ffd7a32d5b.png)
are the feet of the perpendiculars from
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
to
![BC,CA,AB](/media/m/9/8/c/98c204ffa459114826231180fce7ec09.png)
respectively.
Proposed by United Kingdom.
%V0
Find, with proof, the point $P$ in the interior of an acute-angled triangle $ABC$ for which $BL^2+CM^2+AN^2$ is a minimum, where $L,M,N$ are the feet of the perpendiculars from $P$ to $BC,CA,AB$ respectively.
Proposed by United Kingdom.
Izvor: Međunarodna matematička olimpijada, shortlist 1987