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not be a trapezoid such that there is a circle centered at 
that is tangent to the four sides 
, 
, 
, and 
. Let 
, 
, 
, and 
be the circumcenters of the triangles 
, 
, 
, and 
, respectively. Prove that there is a circle containing the circumcenters of the triangles 
, 
, 
, and 
.
and 
for all 
be real numbers such that 
Prove that
be the set of sequences of length 2018 whose terms are in the set 
and sum to 3860. Prove that the cardinality of is at most![\[2^{3860} \cdot \left(\frac{2018}{2048}\right)^{2018}.\]](http://latex.artofproblemsolving.com/d/e/8/de857941803349e275e9c9aa968a201f2988cb8a.png)
where 
denotes the Fibonacci sequence.**
be a sequence such that 
and 
.
and 
)
is a derivable function with the property 
, for every 
, then f is a polynomial function of degree smaller or equal than 1
be a function that admits a primitive 
such that 
if 
such that 
be holomorphic functions on 
, where is an open, connected subset of 
. Suppose that the function 
given by 
takes a maximum value in . Must each function 
be constant on ?
of the triangle 
consider a mobile point 
.
the orthogonal projection of 
on 
.
and 
are such that 
is a paralellogram, find the locus of point 
when 
.