AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
integers where 
. Show that there exists at least three positions 
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 -th position integer in the permutation)
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 
-symmedian with respect to triangle 
and 
as the intersection of lines 
and 
, where all indexes taken cyclically.
, 
, 
, and 
are collinear.
(i know 
well). i want to study so if you guys have some easy or normal problems please send me
and 
. At each step, it is allowed to add to one of the polynomials and to multiply another one by the polynomial 
. Can the polynomials become equal at some point?
of degree . Sasha wants to guess it (knowing ). During a turn, Sasha can name a certain segment ![$[a;b]$](http://latex.artofproblemsolving.com/0/0/6/0066f4b2714f88cdaf883b00f877aa1355b7f9ef.png)
and Maxim will give in response the maximum value of on the segment . Will Sasha be able to guess in a finite number of steps?
and 
Prove that
Let 
and 
Prove that
+1 w
not isosceles, the incircle 
. 
is the intersection of with 
. 
is the projection of 
onto 
. 
cut at the second point 
. 
is the projection of 
onto 
. 
cut at the second point 
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 
, and 
for all integers 
. Prove that 
.
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}} $](http://latex.artofproblemsolving.com/c/d/4/cd481672bee1e87d8c97ac623436e0ec75cfa73a.png)
such that:
holds simultaneously.
![$f(x) = 1 + \frac{1}{x} \in (1, 2].$](http://latex.artofproblemsolving.com/b/1/5/b15c7b9c23c29917af3aba34bf47c8a5869574e8.png)
Now we suppose for a moment 
and perform casework on 
then 
so no solutions. If 
then 
so we must have 
for which we find 
If 
then 
so 
for which we find 
means 
so 
giving 
entries of 
is an integer. Prove that 
are integers.
![\[
v_p(n!) \leq \frac{n}{p - 1}
\]](http://latex.artofproblemsolving.com/5/7/c/57ca4d0a6532fb4b97687a97b9c1fb35ad02b9f2.png)
are given such that 
.
in natural numbers m and n.
Expanding 
of 
yields:
Or equivalently:
Note that:
Checking cases 
yields to solutions 
,