be a quadrilateral such that 
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 
.
passes through the orthocenter of 
intersect at the point 
(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) 
,
?
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 
and 
have equal areas.
Y by GeoKing, Rounak_iitr
Two circles with centers 
and 
intersect at 
and 
. Let 
be the circumcircle of the triangle 
; the circle intersect the circles centered at and at points 
and 
, respectively. The point is inside triangle 
and 
intersects at 
. The point 
on is such that 
. Let 
and 
meet at 
, prove that 
.
Proposed by Amir Parsa Hoseini Nayeri, Iran
and 
intersect at 
and 
. Let 
be the circumcircle of the triangle 
; the circle intersect the circles centered at and at points 
and 
, respectively. The point is inside triangle 
and 
intersects at 
. The point 
on is such that 
. Let 
and 
meet at 
, prove that 
.Proposed by Amir Parsa Hoseini Nayeri, Iran
This post has been edited 1 time. Last edited by NJAX, May 31, 2024, 12:34 PM
,
,
and 
is tangent to 
.
.
be the circumcenter of 
.
Claim: 
is concyclic
Claim: 
is tangent to the circumcircle of 
as 
.
.
be the symmetric point of 
be the symmetric point of 
