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 
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 
.
.
to meet 
.
meets the midpoint of 
.
,
.
with side length of 
.
is selected on the diagonal 
such that 
.
as 
.
Y by Mango247
Let 
be the altitudes of an acute non-isosceles triangle 
. Circumference of the triangles meets that of the triangle 
at point 
(different from 
). Let 
be the midpoint of 
and 
be the intersection point of 
and . Prove that the line of centers the circumferences of the triangles and 
is parallel to the line .
(I. Kachan)
be the altitudes of an acute non-isosceles triangle 
. Circumference of the triangles meets that of the triangle 
at point 
(different from 
). Let 
be the midpoint of 
and 
be the intersection point of 
and . Prove that the line of centers the circumferences of the triangles and 
is parallel to the line .(I. Kachan)
This post has been edited 1 time. Last edited by parmenides51, Dec 27, 2020, 7:01 AM
Reason: author
Reason: author
and 
.
is harmonic, so![\[ -1 = (NH;B_1A_1) \stackrel{C}{=} (KD;AB) \implies DK\cdot DM = DA\cdot DM=DP\cdot DC,\]](http://latex.artofproblemsolving.com/5/e/2/5e237ff2b94b2e408c5ef4cb210288c21243f3e0.png)
which implies 
are concyclic. Hence the radical axis of 
and 
is 
,
.
.
is common for both triangles. So dist
,
,
)
