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 the inequality![\[(X+Y)^2(X^2+Y^2+C)+(1-XY)^2 \ge 0\]](http://latex.artofproblemsolving.com/b/7/7/b77a09cb77ac756308011dcf7f301a3072e90383.png)
holds for all real numbers 
and 
. For which values of and does equality hold for this smallest constant ?
is the solution now let 
where 
we intend to try to find a smaller C
for all 
and 
thus it will become negative forcing the 
such that the inequalityholds for all real numbers and . For which values of and does equality hold for this smallest constant ?
.
works using Sum of Squares, but it isn't working.
implies 
. This works because 
. Equality holds if and only if .
such that the inequalityholds for all real numbers and . For which values of and does equality hold for this smallest constant ?
hence if 
then the inequality holds for all real 
. On the other hand, take 
, where 
, and 
to obtain 
, which is a contradiction.
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.
.
:
,
,
Now since 
,
Equality holds iff 
,
.
But that only works for ![$c\in [-1, 2]$](http://latex.artofproblemsolving.com/9/f/e/9fef62bef0f5491dcc08926b5d04547aa2a0470d.png)
,
holds for 
then