is described on a circle 
,
be 
respectively. A circle with a center 
,
and lines 
.
,
and lines 
.
,
and lines 
,
and lines 
are concurrent.
such that 
is a divisor of 
.
,
and 
.
is prime, 
with 
and 
.
Y by
Let 
be a tetrahedron with 
and let 
be any point inside the triangle 
. Denote respectively by 
the distances from to faces 
, 
, 
. Prove that:
(a)
.
(b)
.
(c)
.
by Nguyen Viet Hung - Hanoi - Vietnam
be a tetrahedron with 
and let 
be any point inside the triangle 
. Denote respectively by 
the distances from to faces 
, 
, 
. Prove that:(a)
.(b)
.(c)
.by Nguyen Viet Hung - Hanoi - Vietnam
This post has been edited 1 time. Last edited by parmenides51, Dec 13, 2021, 1:36 PM
lies in the plane 
we have 
so by AM-GM![\[\frac{d_ad_bd_c}{OA\cdot OB\cdot OC} \le \frac{1}{27} \left( \frac{d_a}{OA}+\frac{d_b}{OB}+\frac{d_c}{OC}\right)^3=\frac 1 {27}.\]](http://latex.artofproblemsolving.com/f/6/f/f6f437a03b80a83dbe59fa0b1b3d098f57bee22a.png)
(c) is easy too. By C-B-S![\[\sum OA\cdot d_a^3 = \left(\sum OA\cdot d_a^3\right)\left(\sum \frac {d_a} {OA}\right) \ge \left(\sum d_a^2\right)^2=OP^4.\]](http://latex.artofproblemsolving.com/e/5/1/e512fc1e20eeca814162b440446ff9c3c1834647.png)
