IMO Shortlist 1970 problem 3
Dodao/la:
arhiva2. travnja 2012. In the tetrahedron
![ABCD,\angle BDC=90^o](/media/m/4/8/a/48a03f2574fc021d553418b7e17c47d5.png)
and the foot of the perpendicular from
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
to
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
is the intersection of the altitudes of
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
. Prove that:
![(AB+BC+CA)^2\le6(AD^2+BD^2+CD^2).](/media/m/2/2/f/22f85e05297d97fb4790d875dd39ee0a.png)
When do we have equality?
%V0
In the tetrahedron $ABCD,\angle BDC=90^o$ and the foot of the perpendicular from $D$ to $ABC$ is the intersection of the altitudes of $ABC$. Prove that: $$(AB+BC+CA)^2\le6(AD^2+BD^2+CD^2).$$ When do we have equality?
Izvor: Međunarodna matematička olimpijada, shortlist 1970