and 
Prove that
be a positive integer. For a set of points in the plane 
distinct points 
connected if the line 
points from 
such that there exists a set 
is connected with exactly 
other points from 
.
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 
.
then prove \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{2}\cdot\frac{abc}{ab+bc+ca}\ge \frac{5}{3}.$$
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 
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 
,
.
.
such that 
.
as 
.
Y by Adventure10, Mango247
Points 
and 
are given on sides 
and 
of rhombus 
, respectively. Line 
intersects line 
in point 
, line 
intersects line in point 
, line 
intersects line 
in point 
and line 
intersects line 
in 
. Prove that triangles 
and 
have equal area
and 
are given on sides 
and 
of rhombus 
, respectively. Line 
intersects line 
in point 
, line 
intersects line in point 
, line 
intersects line 
in point 
and line 
intersects line 
in 
. Prove that triangles 
and 
have equal area
