[/quote]
,
,
be the orthocenter of 
.
at 
at 
.
be the midpoint of 
,
meet 
at 
.
is tangent to 
,
and orthocenter 
be a point on 
.
and 
similarly.
,
.
such that 
.
the point on 
.
is the projection of 
passes through the orthocenter of 
be an acute angled scalene triangle with circumcircle 
.
respectively. Let 
and 
be points on segments 
and 
Prove that 
and 
intersect on the circumcircle of 
.
we consider the last digit different from zero in the decimal expansion of 
.
.
be a quadrilateral such that 
and 
are acute and 
.
,
.
.
be a point on the arc 
(not containing 
and 
.
intersect at point 
.
be the reflection of 
and 
intersect at point 
.
,
,
.
and 
.
.
,
exactly once and finally returns with his last jump to 
jumps be a) 
,
?
and 
be points on 
and 
and 
respectively. Prove that the locus of circumcenter of 
is a line.
, such that 
Prove that :
equality cases : ?
such that ![$$\left(a^{3}+b^{3}+c^{3}-3abc\right)^{2}-\left[a^{3}+b^{3}+c^{3}+3abc-ab(a+b)-bc(b+c)-ca(c+a)\right]^{2}\ge 2k\cdot(a-b)^{2}(b-c)^{2}(c-a)^{2}$$](http://latex.artofproblemsolving.com/d/2/4/d24b46e65a587fbd977467b6e4e31838a0eb57aa.png)
forall 
Y by Adventure10
Let 
be an acute triangle with 
. Prove that there exists a point 
with the following property: whenever two distinct points 
and 
lie in the interior of such that the points 
, 
, , and lie on a circle and
holds, the line
passes through .
be an acute triangle with 
. Prove that there exists a point 
with the following property: whenever two distinct points 
and 
lie in the interior of such that the points 
, 
, , and lie on a circle and
holds, the line
passes through .
are tangent to each other at 
.
.
and is independent of 
)
