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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 
, solve the given equation in 
., the equation has at most 
real solutions.
and 
intersect at two points 
and 
. Let 
be the line tangent to these circles at 
and 
, respectively, so that lies closer to than . Let 
be the line parallel to and passing through the point , with 
on and 
on . Lines 
and 
meet at 
; lines 
and meet at 
; lines 
and meet at 
. Show that 
.
with 
is inscribed in a circle 
. Circles 
and 
touch the lines 
and 
, and their centres lie on the circumference of . Prove that 
lies on a common external tangent to and .
with is inscribed in a circle . Circles and touch the lines and , and their centres lie on the circumference of . Prove that lies on a common external tangent to and .
and lay on internal and external bisectors of 
, namely they lay on midpoints of arcs 
and 
, i.e. on perpendicular bisector of , which is symmetry axis of this two circles and points 
and . So, due to this symmetry if one common tangent () goes through , the other goes through .
