[/quote]
,
,
is fantabulous, then all the elements are fantabulous. Does it follow that the number 
is fantabulous?
and 
.
.
are on the sides 
respectively such that 
.
,
?
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
,
.
are different real numbers. Prove that:
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 
.
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)
.
,
is a perfect square. Find the sum of all possible values of 
Y by
Please solve this problem.
There are n-ants in a rope. The rope is 10 meters long. The velocity of each ants is 1 meter/second. If any 2 ants are touched in the rope, they will inverse their way. If any ants reaches in any corner of the rope, it will fell down. Prove that all ants will fell down in 10 seconds.
There are n-ants in a rope. The rope is 10 meters long. The velocity of each ants is 1 meter/second. If any 2 ants are touched in the rope, they will inverse their way. If any ants reaches in any corner of the rope, it will fell down. Prove that all ants will fell down in 10 seconds.
