and 
.
Equality holds when 
or 
and 
,
cookies from one of the piles, of which he eats 
and puts 
for which Marica has a winning strategy.
be non-negative real numbers such that 
Prove that
is the incentre and point 
is defined such that 
and 
is the perpendicular bisector of 
and 
be the circles passing through 
and 
respectively such that 
is tangent to 
to be a point such that it lies on both the circles 
and 
are perpendicular.
weights, each weighing 
and another 
weights with 
each. Write a algorithm that determines the ways in which a scale can be balanced with a weight 
on the left pan, and display the number of possible solutions. (The weights can be placed on both pans and the program starts with the numbers 
.
with orthocenter 
.
to meet 
.
meets the midpoint of 
.
,
.
with side length of 
.
is selected on the diagonal 
such that 
.
as 
.
.
as 
respectively, compute the area of 
given the area of 
.
.
lies inside the rectangle such that 
is equilateral. Given that 
and the midpoint of 
a
.
Y by Adventure10, Mango247
Let 
be a triangle. Let 
denote the reflection of 
in the internal angle bisector 
of 
.Show that the circumcentre of the triangle 
lies on the line where 
is the incentre of .
be a triangle. Let 
denote the reflection of 
in the internal angle bisector 
of 
.Show that the circumcentre of the triangle 
lies on the line where 
is the incentre of .This post has been edited 1 time. Last edited by parmenides51, Sep 30, 2018, 1:07 AM
Reason: name edit
Reason: name edit
![\[\angle DB'A = \angle ABD = \angle DBC\]](http://latex.artofproblemsolving.com/4/f/5/4f599c4edbdb213fdfd71d237db3e294beafbb21.png)
and 
are concyclic. We know that the circumcenter of 
is the midpoint of arc 
of 
.
