[/quote]
,
the point 
is the centre of the excircle opposite the vertex 
This excircle is tangent to the side 
at 
,
and 
at 
and 
,
and 
meet at 
,
and 
meet at 
Let 
be the point of intersection of the lines 
and 
be the point of intersection of the lines 
and 
Prove that 
is the circle that is tangent to the line segment 
,
.
such that if 
and 
are distinct rational numbers satisfying 
or 
,
?
and 
.
be a positive integer. Consider the following equation: ![\[ \{x\}+\{2x\}+ \dots + \{nx\} = \lfloor x \rfloor + \lfloor 2x \rfloor + \dots + \lfloor 2nx \rfloor\]](http://latex.artofproblemsolving.com/8/2/9/8299b3ceb1650418c0a373536cb1d2931900efd9.png)
a) For 
,
.
real solutions.
and 
intersect at two points 
and 
.
be the line tangent to these circles at 
and 
,
be the line parallel to 
on 
on 
and 
meet at 
;
and 
;
and 
.
.
is such that 
for any positive integers 
.
such that 
for all positive integers 
.
.
satisfy that 
.
with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of 
be real numbers . Prove that : There exist positive integer 
such that 
Where 
)
.
passes through 
.
and 
,
be the midpoint of 
.
,
and 
Y by
Given two charges of any magnitude , a third charge collinear with them , exists such that it is in equillibirum , Prove that if a fourth charge in the same line exists such that it is in equillibrium , then the 3rd charge and the fourth charge are harmonic conjugates with respect to the two fixed charges . , For example if two +q charges are fixed then if in their midpoint placed a charge -q , it is in equillibrium , also if the same charge -q is placed at infinity the system is again in equillibrium , and the midpoint and the point at infinity are harmonic conjugates .
