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 
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 
.
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 the center of the circumcircle of an acute triangle 
, let 
be any point inside the segment 
. Suppose the circumcircle of triangle 
intersects the segment 
at point 
and the circumcircle of triangle 
intersects 
at point 
.
(i) Consider the triangle
, show that it is similar to triangle and that is its orthocenter.
(ii) Show that the circumcircles of triangles, , have the same radius.
be the center of the circumcircle of an acute triangle 
, let 
be any point inside the segment 
. Suppose the circumcircle of triangle 
intersects the segment 
at point 
and the circumcircle of triangle 
intersects 
at point 
.(i) Consider the triangle
, show that it is similar to triangle and that is its orthocenter.(ii) Show that the circumcircles of triangles
, , have the same radius.
.
,
.
,
be the circumradius of 
,
,
,
and 
(say 
)
as desired. 
