be a point in the plane, distinct from the vertices of 
.
the reflections of 
and 
,
are collinear if and only if 
coincide, prove that 
such that ![\[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]](http://latex.artofproblemsolving.com/0/6/2/0623b7c328af6d088cfc5dd288afe3e17a94ff29.png)
for all reals 
and 
.
be a fixed positive integer, and 
be a sequence of nonnegative real numbers such that, for all 
,
.
is bounded above.
is bounded above.
are positiv real number, such that 
prove inequality 
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
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 
.
Y by Adventure10, Mango247
Let 
be an acute angled triangle, and 
a point in its interior. Let the reflections of through the sides 
and 
be called 
and 
, respectively, and let the reflections of H through the midpoints of these same sidesbe called 
and 
, respectively. Show that the four points 
, and are concyclic if and only if at least two of them coincide or lies on the altitude from 
in triangle .
be an acute angled triangle, and 
a point in its interior. Let the reflections of through the sides 
and 
be called 
and 
, respectively, and let the reflections of H through the midpoints of these same sidesbe called 
and 
, respectively. Show that the four points 
, and are concyclic if and only if at least two of them coincide or lies on the altitude from 
in triangle .
,
and 
.
sends 
to 
,
are concyclic 
.
