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 
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 
.
with orthocenter 
.
to meet 
.
meets the midpoint of 
.
,
.
with side length of 
.
is selected on the diagonal 
.
as 
.
Y by
Let 
and 
be points on side 
of a triangle 
. Points 
and 
are chosen on segments 
and 
, respectively, so that 
and 
. The incircles of 
and 
touches segments and at 
and 
, respectively. Lines and 
intersect at 
, and lines and 
intersect at 
. Given that 
, prove that 
.
and 
be points on side 
of a triangle 
. Points 
and 
are chosen on segments 
and 
, respectively, so that 
and 
. The incircles of 
and 
touches segments and at 
and 
, respectively. Lines and 
intersect at 
, and lines and 
intersect at 
. Given that 
, prove that 
.This post has been edited 1 time. Last edited by parmenides51, Dec 18, 2020, 4:53 AM
Reason: typo as mentioned in#2
Reason: typo as mentioned in#2
and 
and 
,
touches 
at 
, 
, thus ![\[\frac{AM}{AN}=\frac{KE}{KX}\cdot \frac{LY}{LF}=\frac{KE}{KB}=\frac{AC}{AB}\]](http://latex.artofproblemsolving.com/6/0/a/60ad6bd39a1f38dbef173785126e358999035bb4.png)
as desired. 
