[/quote]
,
we consider the last digit different from zero in the decimal expansion of 
.
.
and 
are acute and 
.
,
.
be circumscribed about circle 
,
be the orthocenter of 
.
at 
.
at 
.
be the midpoint of 
,
meet 
at 
.
is tangent to 
.
such that 
.
the point on 
.
is the projection of 
passes through the orthocenter of 
and 
intersect at the point 
.
be a point on the arc 
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 
board, show that there is a 
block on the board in which at least 3 of the squares are colored red.
and let 
and 
and 
have equal areas.
Y by Maths_Guy, Adventure10, Mango247, Rounak_iitr
In a cyclic quadrilateral 
with circumcenter 
, the diagonals 
and 
intersect at 
. Let the circumcircles of triangles 
and 
intersect at 
. Let the circumcircles of triangles 
and 
intersect at 
. If lies inside and if the points 
are all distinct, prove that lie on a circle.
with circumcenter 
, the diagonals 
and 
intersect at 
. Let the circumcircles of triangles 
and 
intersect at 
. Let the circumcircles of triangles 
and 
intersect at 
. If lies inside and if the points 
are all distinct, prove that lie on a circle.This post has been edited 1 time. Last edited by TheDarkPrince, Oct 29, 2018, 3:56 PM
,
,
and 
,
.
,
lie on a circle. QED
are concyclic.
.
.
and 
is the angle bisector of 
,
.
and we are done.
in place of 
are cyclic quadrilaterals, now invert around 
and (denote the image of any point 
)
can anyone please explain.
.
are concyclic
to get:
are concylic
, hence we get points 
hence we get that points 
.
,
.
as well, giving that 
and 
,
