and 
Prove that
be a positive integer. For a set of points in the plane 
,
distinct points 
connected if the line 
contains exactly 
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 
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 
.
Y by Adventure10, Mango247
Circles 
and 
touch the circle 
in an inner way in points 
and 
correspondingly. It is also known that the circles 
and 
touch each other in an outer way in the pont 
, circles and touch each other in an outer way in the pont 
, and circles and touch each other in an outer way in the pont 
. Prove that straight lines 
and 
ave a common point.
(O. Manzjuk, Kyiv)
and 
touch the circle 
in an inner way in points 
and 
correspondingly. It is also known that the circles 
and 
touch each other in an outer way in the pont 
, circles and touch each other in an outer way in the pont 
, and circles and touch each other in an outer way in the pont 
. Prove that straight lines 
and 
ave a common point.(O. Manzjuk, Kyiv)
