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 
and 
is the perpendicular bisector of 
and 
be the circles passing through 
and 
respectively such that 
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 
,
.
.
is selected on the diagonal 
.
as 
.
.
as 
respectively, compute the area of 
given the area of 
.
.
is equilateral. Given that 
and the midpoint of 
a
.
Y by Adventure10, Mango247
Let 
be a cyclic quadrilaterla and let 
be the point of intersection of its diagonals 
and 
. Suppose 
and 
meet in 
. Let the midpoints of 
and 
be 
and 
respectively. If 
is the circumcircle of triangle 
, prove that 
is tangent to .
be a cyclic quadrilaterla and let 
be the point of intersection of its diagonals 
and 
. Suppose 
and 
meet in 
. Let the midpoints of 
and 
be 
and 
respectively. If 
is the circumcircle of triangle 
, prove that 
is tangent to .
with power 
,
are interchanged and also 
interchange. Suppose, 
goes to 
.
.
.
.
. So in this inversion 
go to 
respectively. So 
is anti-parallel to 
wrt angle-bisector of 
.
because angle-bisectors of 
.
is anti-parallel to 
.
cyclic let us produce 
to meet 
.
.
is tangent to 
.
to 
.
.
,
be the transformation consisting of a reflection about the bisector of 
and a dilation with factor 
,
be the same transformation, but with a factor of 
instead.
,
.
,
.
.
are collinear and 
.
