[/quote]
,
and 
be positive integers with 
such that 
divides 
.
and 
;
and 
.
with their incenter 
.
lies on 
on the same side as 
wrt 
.
lies on the shorter arc 
of the circumcircle 
Prove that 
is the angle bisector of 
.
.
represented by tokens to players 
to 
so that they all have the same number of tokens in their urns.
beats 
when, when 
.
What is 
such that the following equations are both true:
be reals such that 
Prove that
of the triangle 
.
intersect 
and denote the excentre of 
.
and 
be the midpoints of 
respectively.
and 
are concyclic.
.
.
cut 
at 
and 
be the centroid of 
meets 
and 
is the projection of 
onto 
lie on a common circle.
be 
.
and 
are two moving points on the circle 
:
,
.
.
.
such that there exists a non-negative integer 
for which the set 
is fragrant?
and 
Show that
Where 
Where 
is a positive integer, then a user is an 
-
-
such that the equation![\[f(xf(x)+y) = f(y) + x^2\]](http://latex.artofproblemsolving.com/2/0/a/20a245460634f0455fc703d7ce41c475c1ae4970.png)
holds for all rational numbers 
and 
.
denotes the set of rational numbers.
Y by Adventure10, Mango247
We say that a finite set 
of red and green points in the plane is separable if there exists a triangle 
such that all points of one colour lie strictly inside and all points of the other colour lie strictly outside of . Let 
be a finite set of red and green points in the plane, in general position. Is it always true that if every 
points in form a separable set then is also separable?
of red and green points in the plane is separable if there exists a triangle 
such that all points of one colour lie strictly inside and all points of the other colour lie strictly outside of . Let 
be a finite set of red and green points in the plane, in general position. Is it always true that if every 
points in form a separable set then is also separable?
with center 
.
,
is sufficiently small.
