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
Let 
be a polyhedron whose all edges are congruent and tangent to a sphere. Suppose that one of the facesof has an odd number of sides. Prove that all vertices of lie on a single sphere.
be a polyhedron whose all edges are congruent and tangent to a sphere. Suppose that one of the facesof has an odd number of sides. Prove that all vertices of lie on a single sphere.
be the center of the sphere to which the edges are tangent, and let 
be its radius. Let 
be the side length of an isosceles triangle whose base is the length of an edge of 
of 
is less than, greater than or equal to 
