AoPS Academy
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plane, given a parabola 
passing through 3 points 
and 
.
vary in the range of 
, draw the domain swept out by .
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
, where![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
is defined by 
and 
Show that there exist infinitely many prime number that divide at least one number in this sequences
be an acute scalene triangle , 
be the orthogonal projection of 
on 
, 
and 
are the midpoints of 
and 
respectively. Let 
and 
are points on the minor arcs 
and 
of the circumcircle of triangle respectively such that 
. Show that the circumcircles of triangles 
and 
are tangent if and only if lies on 
.
be the side lengths of an non-degenerate triangle with 
. With 
denote the minimum of the quotients 
and 
. Find all values that can take.
be the side lengths of an non-degenerate triangle with . With denote the minimum of the quotients and . Find all values that can take.
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or 
results 
.
and 
we consider 
cases:
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,
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is the interval
.
