[/quote]
,
is an integer. Prove that 
are integers.
integers where 
.
such that ![$gcd(p[i], p[i+1])=1$](http://latex.artofproblemsolving.com/f/b/b/fbb304fb8ebdb8987edc830234698de2dc7a28a2.png)
(here ![$p[i]$](http://latex.artofproblemsolving.com/0/b/6/0b6cc7e1108ab6b64f3a644a3b28891f826b3bea.png)
refers to the 
be not cyclic and haves edges not parallel to each other.
as the intersection of the tangent line at 
with respect to circle 
and the 
-
and 
as the intersection of lines 
and 
,
,
,
,
are collinear.
(
well). i want to study so if you guys have some easy or normal problems please send me
and 
.
.
of degree ![$[a;b]$](http://latex.artofproblemsolving.com/0/0/6/0066f4b2714f88cdaf883b00f877aa1355b7f9ef.png)
and Maxim will give in response the maximum value of 
![\[
v_p(n!) \leq \frac{n}{p - 1}
\]](http://latex.artofproblemsolving.com/5/7/c/57ca4d0a6532fb4b97687a97b9c1fb35ad02b9f2.png)
and 
Prove that
Let 
and 
Prove that
not isosceles, the incircle 
.
is the intersection of 
.
is the projection of 
onto 
.
cut 
.
is the projection of 
.
cut 
respectively. Prove 
are collinear
be the set of non-negative integers and 
be the set of positive real numbers. Let 
be a function such that 
and 
for all integers 
,
for all integers 
.
.
which satisfy 
![$ \frac{a+b+c}{3} \ge \sqrt[10]{\frac{a^3+b^3+c^3}{3}}$](http://latex.artofproblemsolving.com/c/d/a/cdae05ad34239366b0dd3921243522c711b6ae58.png)
method, I can assum 
![$ \frac{2x+\frac{1}{x^2}}{3} \ge \sqrt[10]{ \frac{2x^3+ \frac{1}{x^6}}{3}} $](http://latex.artofproblemsolving.com/c/d/4/cd481672bee1e87d8c97ac623436e0ec75cfa73a.png)
are given such that 
.
in natural numbers m and n.
Y by
Sabbir noticed one day that everyone in the city of BdMO has a distinct word of length 
, where each letter is either 
or 
. Sabbir saw that two citizens are friends if one of their words can be altered a few times using a special rule and transformed into the other ones word. The rule is, if somewhere in the word 
is located consecutively, then these letters can be changed to 
or if is located somewhere in the word consecutively, then these letters can be changed to (if wanted, the word can be kept as it is, without making this change.) For example 
can be transformed into 
(the opposite is also possible.) Now Sabbir made a team of 
citizens where no one is friends with anyone. What is the highest value of 
, where each letter is either 
or 
. Sabbir saw that two citizens are friends if one of their words can be altered a few times using a special rule and transformed into the other ones word. The rule is, if somewhere in the word 
is located consecutively, then these letters can be changed to 
or if is located somewhere in the word consecutively, then these letters can be changed to (if wanted, the word can be kept as it is, without making this change.) For example 
can be transformed into 
(the opposite is also possible.) Now Sabbir made a team of 
citizens where no one is friends with anyone. What is the highest value of 
