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 
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 
.
,
.
.
such that 
.
as 
.
Y by Adventure10, Mango247
Diagonals of a convex quadrilateral 
are equal and intersect at a point 
. Inside the triangles 
and 
select points 
and 
so that 
Prove that line 
is parallel to the bisector of the angle .
are equal and intersect at a point 
. Inside the triangles 
and 
select points 
and 
so that 
Prove that line 
is parallel to the bisector of the angle .This post has been edited 1 time. Last edited by parmenides51, Sep 9, 2018, 7:01 PM
Reason: no edit
Reason: no edit
