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and 
.
satisfies the following three conditions:
All elements of are a subset of 
.
.
is the foot of the altitude from 
of triangle 
. On the lines 
and 
points 
and 
are marked such that the circumcircles of triangles 
and 
are tangent, call this circles 
and 
respectively. Tangent lines to circles and at and intersect at 
.
.
be non-negative real numbers such that 
Prove that
, sequence 
satisfies the equation:![$\sum_{k=1}^{n}\frac{1}{2}(1-(-1)^{[\frac{n}{k}]})a_{k}=1$](http://latex.artofproblemsolving.com/3/2/1/3216129b3e42dfb49eb8bb764fafa1b9fbf28882.png)
, find the value of 
. such that there exist two permutations 
and 
of the set 
, satisfying the condition
for all 
.
with 
. Let 
be a point of 
such that 
is tangent to the circumcircle of . Let 
and 
be the circumcenters of triangles 
and 
, respectively, and let 
be the midpoints 
. Prove that the line tangent to the circumcircle of 
through is also tangent to the circumcircle of .
such that for all 
, we have 
.
be a function over reals sich that 
is surjective.
for all reals 
, what is the largest possible value of |V| if |E|=35 and 
such that for any two infinite sequences of positive integers 
and 
, such that 
for all 
, there always exists a real number 
such that 
for all 
.
)
is the circumcenter, the 
i
and 
are perpendicular. Prove that 
and 
