[/quote]
,
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 
be an isosceles triangle 
)
.
passes through 
and 
.
and circumcircle of 
and 
,
be the midpoint of 
and 
be the midpoint of 
.
,
and 
are concurrent.
Y by PikaPika999
9Poll:
How was the difficulty of the problems compared to last year?
168 Votes
59%
(99)
25%
(42)
4%
(7)
5%
(9)
2%
(4)
4%
(7)
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I personally thought the problems were much easier than last year, but I didn't really improve as much as I would of liked to improve.
