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with circumcenter 
and orthocenter 
, let 
be an arbitrary point on the circumcircle of triangle such that does not lie on line 
and that line 
is not parallel to line 
. Let 
be the point on the circumcircle of triangle such that 
is perpendicular to , and let 
be the point on line 
such that 
. Let 
and 
be the points on the circumcircle of triangle such that 
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)
. If this is still too hard assume in 
that 
is a prime.
sastifying 
are there such that 
and 
?
, where 
is a real number. Prove that for all positive , 
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 and 
Prove that
Let and 
Prove that
Let and 
Prove that
let 
and 
be the circles passing through 
and 
respectively such that is tangent to and define to be a point such that it lies on both the circles and prove 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 
. What will be displayed after three successive runs: 5,2,5,1,4 | 5,2,5,1,11 | 5,2,5,1,20?
on a circle 
. Find the locus of the incenters of all triangles when point 
describes .
of the triangle is the intersection of the lines 
, 
, where and 
are the midpoints of the arcs 
and not containing 
and 
respectively.
of the circle. Let 
be the angle measure of the minor arc 
, and 
the measure of the major arc.
.
. Since this is constant, we notice that the locus of all points I is the arc of a circle with chord where chord corresponds to an arc of measure 
.
. in this case is the arc of a circle with chord where chord corresponds to an arc of measure 
. are the locus of all possible locations of points .
.
,
,
,
.