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
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)
be an odd integer. In a 
board, we colour 
cells. What is the largest number of three-square corners that can surely be cut out of the uncoloured figure?
.
we have
we use induction.
parts. Let there be 
times 
squares which contain black squares.![\[2(n-1)^2=f(1)+2f(2)+3f(3)+4f(4)\geq f(1)+2(f(2)+f(3)+f(4))=f(1)+2(n^2-f(0)-f(1))=2n^2-2f(0)-f(1)\]](http://latex.artofproblemsolving.com/e/c/9/ec92c9bf613bbde1ffd9550f4db0c6ca42693d78.png)
![\[2f(0)+2f(1)\geq 2f(0)+f(1)\geq 4n-2\implies f(0)+f(1)\geq 2n-1\]](http://latex.artofproblemsolving.com/1/5/d/15ddcd111a0862bbb9fdb4e278892840497ee658.png)
Thus there must exist at least times squares which can contain 
square corner as desired.
. the following construction can be verified to work![[asy]
filldraw(((1,2)--(1,4)--(2,4)--(2,2)--cycle), gray);
filldraw(((4,2)--(4,4)--(5,4)--(5,2)--cycle), gray);
filldraw(((2,1)--(4,1)--(4,2)--(2,2)--cycle), gray);
filldraw(((2,4)--(4,4)--(4,5)--(2,5)--cycle), gray);
add(grid(6,6));
[/asy]](http://latex.artofproblemsolving.com/3/e/9/3e9a7d6b687aacf4239543f7a8b6b65904e8bbe1.png)
place a copy for the 
construction for in the center of the 
grid. Then with the additional 
squares surround the central construction. Then you are able to cut out 
additional three corner pieces from the corners in addition to the three corners from the center. squares. In order to stop a three corner from being cut out from one of these squares there must be at least two squares shaded. Notice there can only by 
such squares with at least two squares shaded. Thus we can cut out three corners form at least 
of them.
without one (any) cell.
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.
