AoPS Academy
. Show that there exists 
such that![\[a_1x_1^2+a_2x_2^2+\ldots +a_nx_n^2\geqslant (a_1x_1+a_2x_2+\ldots +a_nx_n)^2.\]](http://latex.artofproblemsolving.com/4/d/2/4d269e5b682be6778e89237aa91da5e0399042a7.png)
, inscribed in the circle 
, satisfies 
and 
, and 
is the intersection point of the diagonals 
and 
. The circle with center 
and radius 
intersects in two points 
and 
. Prove that the line 
is tangent to the circles with diameters 
and 
.
be a tetrahedron such that 
, 
be the radius of its inscribed sphere and 
be the orthocenter of triangle 
. Prove that 
be a tetrahedron such that 
, 
, 
and be an orthocenter of the triangle .![$$OH\leq\frac{\sqrt[3]{HA\cdot HB\cdot HC}}{\sqrt2}.$$](http://latex.artofproblemsolving.com/c/9/f/c9fa62742e000e3e52308c61006371b6a4aea8af.png)
be a tetrahedron such that , , and be an orthocenter of the triangle .
be a tetrahedron such that , , and be an orthocenter of the triangle . Then foot of 
on plane is orthocenter of triangle .
where 
.
be a tetrahedron such that , , and be an orthocenter of the triangle .
satisfying the equation![\[
(f(x+y))^2= f(x^2) + f(2xf(y) + y^2), \quad \forall x, y \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/2/5/e/25eed911f303ed7cc4e03765a2f943ad4641c451.png)
plane 
is perpendicular to line 
.
.
hence plane 
is perpendicular to line 
.
.
is perpendicular to plane