are positiv real number, such that 
prove inequality 
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.
be a parallelogram. Let line 
externally bisect 
and let 
be the line passing through 
which is parallel to line 
.
at point 
and 
,
at point 
.
meet line 
at point 
and the internal bisector of 
at point 
.
,
are concyclic.
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
is the incentre and point 
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 
.
with orthocenter 
.
to meet 
meets the midpoint of 
.
,
.
.
is selected on the diagonal 
.
as 
.
.
as 
respectively, compute the area of 
given the area of 
.
Y by Adventure10, Mango247
In the triangle 
, the sides 
and 
are equal. The point 
inside the triangle is such that the angle 
is twice as large as the angle . Prove that the double distance from the point 
to the line bisecting the angles externally to the angle is 
.
, the sides 
and 
are equal. The point 
inside the triangle is such that the angle 
is twice as large as the angle . Prove that the double distance from the point 
to the line bisecting the angles externally to the angle is 
.
and 
and let 
to 
,
is circumcenter WRT 
