for which, for all 
,![\[
f(x) f(yf(x)) + y f(xy) = \frac{f\left(\frac{x}{y}\right)}{y} + \frac{f\left(\frac{y}{x}\right)}{x}
\]](http://latex.artofproblemsolving.com/9/0/c/90c180110402e1a32b70edb2b0a03a28727457d1.png)
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
,![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
be a 
on it, line 
makes a fixed angle with the tangent to 
be a rectangular hyperbola centered at the origin, scaled such that it is tangent to the logarithmic spiral at some point.
on the spiral, the polar of 
Y by Adventure10, Mango247
Assume 
are three collinear points that ![$B \in [AC]$](//latex.artofproblemsolving.com/a/d/9/ad95e90f7c54f31cf8424b01e8d0c2f9e79e5f6c.png)
. Suppose 
and 
are to parrallel lines that
, 
and 
are not collinear. Suppose 
is circumcenter of circle passing through 
, and . Also 
is circumcenter of circle passing through 
, and . If area of 
is equal to area of 
, then find all possible values for 
are three collinear points that ![$B \in [AC]$](http://latex.artofproblemsolving.com/a/d/9/ad95e90f7c54f31cf8424b01e8d0c2f9e79e5f6c.png)
. Suppose 
and 
are to parrallel lines that
, 
and 
are not collinear. Suppose 
is circumcenter of circle passing through 
, and . Also 
is circumcenter of circle passing through 
, and . If area of 
is equal to area of 
, then find all possible values for 
be the intersection of the circles 
other than the point 
of these 2 circles is perpendicular to their center line 
and the center line cuts the segment 
.
are parallel, the angles 
are equal. Thus the angles spanning the major arcs 
of the circles 
are equal, which means that the points 
are collinear. The triangles 
are both isosceles. Since 
,
,
span the major arcs 
and the corresponding external angles 
of the triangles 
are equal, which means that these 2 triangles are always similar, regardless of the angle 
are congruent, iff their corresponding sides 
are equal. The segment 
is the circumradius of the triangle 
and the segment 
its side against the angle 
.
or 
.
