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 defined such that 
and 
is the perpendicular bisector of 
and 
be the circles passing through 
and 
respectively such that 
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 
.
.
to meet 
.
meets the midpoint of 
.
,
.
with side length of 
.
is selected on the diagonal 
.
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
Circle with center 
, inscribed in a triangle 
, touches the sides 
and 
at points 
and 
respectively. On rays 
and 
, respectively, let be the points 
and 
such that 
, where 
is the radius of the circumscribed circle of the triangle . Prove that:
a)
where 
is the center of the circumscribed circle of the triangle ,
b) lines
and 
intersect on the circumcircle of the triangle .
, inscribed in a triangle 
, touches the sides 
and 
at points 
and 
respectively. On rays 
and 
, respectively, let be the points 
and 
such that 
, where 
is the radius of the circumscribed circle of the triangle . Prove that:a)
where 
is the center of the circumscribed circle of the triangle ,b) lines
and 
intersect on the circumcircle of the triangle .
