[/quote]
,
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)
,
is fantabulous, then all the elements are fantabulous. Does it follow that the number 
is fantabulous?
and 
.
that satisfies:![\[
\sum_{i=1}^{n} a_{\left\lfloor \frac{n}{i} \right\rfloor} = n^k
\]](http://latex.artofproblemsolving.com/b/b/e/bbe7fe81e366fe390be5d17193d1581cc40dbece.png)
be a positive integer. Prove that for all integers 
,![\[
\frac{c^{a_n} - c^{a_{n-1}}}{n} \in \mathbb{Z}
\]](http://latex.artofproblemsolving.com/0/4/4/04464c1a350e3f9a929ab51534018498680068fc.png)
such that![\[
f(f(x)+y) = x+f(f(y))
\]](http://latex.artofproblemsolving.com/9/d/7/9d780faa5901646f7035e88400a9e3d8023f04ff.png)
for all 
.
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle 
with circumcenter 
,
and orthocenter 
.
is constant.
-
passing through 3 points 
and 
.
vary in the range of 
,
is defined by 
and 
Show that there exist infinitely many prime number that divide at least one number in this sequences
on 
and 
are the midpoints of 
and 
are points on the minor arcs 
and 
of the circumcircle of triangle 
.
and 
are tangent if and only if 
.
be 
given points in the plane, and let 
be a real number. Alice and Bob play the following game. Firstly, Alice constructs a connected graph with vertices at the points 
a non-negative real number 
,
,
.
such that 
and 
are connected by an edge for every 
.
is not fixed and the first selected vertex always has to be 
.![\[
\frac{1}{k+1} \sum_{j=0}^{k} r_{i_j} \geq r;
\]](http://latex.artofproblemsolving.com/1/7/9/1795db8e4a509dd465f6ff462093ae75b04de2b4.png)
otherwise, Alice wins. Depending on 
,
for which Bob has a winning strategy.
such that![\[
f(x^2 \cdot f(x) + f(y)) = f(f(x^3)) + y
\]](http://latex.artofproblemsolving.com/8/6/5/865b10bbc5ba7721078b1c60eaeede2abfeeb1a2.png)
.
.
When does equality hold?
,
is the incenter, 
such that 
and 
.
.
is tangent to 
and 
is tangent to 
,
and 
and 
.
and 
.
as the intersections of 
and 
with lines 
and 
,
is the intersection of 
and 
.
Y by Adventure10, Mango247
Let 
be a right-angled triangle with 
and circumcircle 
. The inscribed circle is tangent to 
in point 
. Let 
be the midpoint of the arc 
of not containing 
and let 
be the midpoint of the arc 
of not containing 
.
(a) Prove that
.
(b) Prove that
goes through the points of tangency of the incircle to and .
be a right-angled triangle with 
and circumcircle 
. The inscribed circle is tangent to 
in point 
. Let 
be the midpoint of the arc 
of not containing 
and let 
be the midpoint of the arc 
of not containing 
.(a) Prove that
.(b) Prove that
goes through the points of tangency of the incircle to and .
with 
or 
.
is cyclic and
.
Since 
we have that 
is cyclic. So
are colinear and similarly we show the other part.
