[/quote]
,
,
be the orthocenter of it and 
be any point on the side 
.
are on the segments 
,
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
.
and 
such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
be the set of all lines in the plane and 
points in the plane. For a line 
let 
be the number of distinct points among the orthogonal projections of the points from 
.
,
,
,
meet the sides 
,
at points 
,
,
be the incenter of triangle 
,
be the foot of the perpendicular from 
.
is cyclic, then the center of its circumcircle lies on segment 
.
be an interior point for it such that 
.
When does equality hold?
,
,
and circumcentre 
.
be the midpoint of 
and let 
be a point such that 
is a parallellogram. Suppose that there exists a point 
on 
to 
.
be the midpoint of 
.
,
.
are the midpoints of 
and 
and 
of the circumcircle of triangle 
.
and 
are tangent if and only if 
.
and 
on 
and 
.
with 
and 
are 
and 
respectively, and the intersections of 
with 
and 
are 
,
and 
are concurrent.
be fixed points on this order on a line. Let 
be a variable circle through 
and 
at the points 
and 
and 
with 
passes through a fixed point independent of 
,
,
satisfying 
?
the sum of all positive divisors of 
be an integer. Alice and Bob play the following game on the vertices of a regular 
rounds: In Alice’s turn on the 
-
position clockwise or anticlockwise. If at the end of any person’s turn the two tokens are on the same vertex, then Alice wins the game, otherwise Bob wins. Decide for each value of 
is not a perfect cube of an integer.
such that if 
denotes the number of positive integers less than or equal to 
be an integer and denote 
.
we denote Im 
,
and 
.
I
+ 
.
Y by Adventure10, Mango247, Rounak_iitr
Let 
be a triangle and 
a point on the internal angle bisector of 
. Circle 
is circumscribed to triangle 
and intersects the segment 
in point 
. Circle 
is circumscribed to the triangle 
. Radius of the cirlce is larger than the radius of . Circle centered at with radius 
intersects the circle in points 
and 
. Circle centered at with radius 
intersects in points 
and 
. Prove 
.
Proposed by Matija Bucić.
be a triangle and 
a point on the internal angle bisector of 
. Circle 
is circumscribed to triangle 
and intersects the segment 
in point 
. Circle 
is circumscribed to the triangle 
. Radius of the cirlce is larger than the radius of . Circle centered at with radius 
intersects the circle in points 
and 
. Circle centered at with radius 
intersects in points 
and 
. Prove 
.Proposed by Matija Bucić.
and 
give 
,
?
