with circumcenter 
and orthocenter 
,
be an arbitrary point on the circumcircle of triangle 
and that line 
is not parallel to line 
.
be the point on the circumcircle of triangle 
is perpendicular to 
be the point on line 
such that 
.
and 
be the points on the circumcircle of triangle 
is a diameter, and 
and 
are parallel. Let 
be the midpoint of 
.
and 
are perpendicular.
and 
are perpendicular.
such that![\[(a^2-b^2)(b^2-c^2) = abc.\]](http://latex.artofproblemsolving.com/7/1/8/7188261c40a45f5e15533100432d36abbf515efd.png)
.
that 
is a prime.
sastifying 
are there such that 
and 
?
, where 
is a real number. Prove that for all positive 
then prove
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
let 
and 
be the circles passing through 
and 
respectively such that 
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 
.
Y by Adventure10, Mango247
On the coordinate plane 
, a parabola 
is drawn. Let 
, 
and 
be different points of this parabola. We define the point 
as the intersection point of the line 
and the 
axis. Similarly, we define the points 
and 
. Prove that the sum of the distance from , and to the 
axis is greater than the sum of the distance from , and to the axis.
, a parabola 
is drawn. Let 
, 
and 
be different points of this parabola. We define the point 
as the intersection point of the line 
and the 
axis. Similarly, we define the points 
and 
. Prove that the sum of the distance from , and to the 
axis is greater than the sum of the distance from , and to the axis.
intersects 
axis in 
and the sum of the distance from 
so first sum is greater
