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)
are taken on the semicircle with diameter 
(points 
are in the specified order), and points 
on the diameter 
)
and 
.
.
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 
be a non-degenerate tetrahedron with 
,
,
,
Let a sphere of radius 
be circumscribed about this tetrahedron. Prove that ![$$r^2+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge 9\sqrt[3]{16}.$$](http://latex.artofproblemsolving.com/8/a/4/8a4a6595e2e5f20222f6288f89d6938301c9d7d2.png)
be a permutation of 
such that there are exactly 
ordered pairs of integers 
satisfying 
and 
.
:
such that 
.
Where 
with legs 
and 
is drawn with a semicircle inscribed into the triangle. What is the smallest possible radius of the semi-circle and the largest possible radius?
be non-negative real numbers such that 
.![\[
P = \frac{a}{a^2 + 2} + \frac{b}{b^2 + 2} + \frac{c}{c^2 + 2}.
\]](http://latex.artofproblemsolving.com/d/b/e/dbea7db769c4996fc081ee75837e177b298cffcb.png)
Y by farhad.fritl, cubres
An infinite increasing sequence 
of positive integers is called central if for every positive integer 
, the arithmetic mean of the first 
terms of the sequence is equal to . Show that there exists an infinite sequence 
of positive integers such that for every central sequence 
, there are infinitely many positive integers with 
.
of positive integers is called central if for every positive integer 
, the arithmetic mean of the first 
terms of the sequence is equal to . Show that there exists an infinite sequence 
of positive integers such that for every central sequence 
, there are infinitely many positive integers with 
.
