[/quote]
,
is an integer. Prove that 
are integers.
.
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 
onto 
.
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 HWenslawski
On a board, Ana writes 
different integers, while Ben writes 
different integers. Then, Ana adds each of her numbers with with each of Ben’s numbers and she obtains 
different integers. On the other hand, Ben substracts each of his numbers from each of Ana’s numbers and he gets 
different integers.
For each integer
, let 
be the number of ways that may be written as sum of one number of Ana and one number of Ben.
a) Show that there exist an integer such that,
b) Does there exist an integer such that,
Proposed by Besfort Shala, Kosovo
different integers, while Ben writes 
different integers. Then, Ana adds each of her numbers with with each of Ben’s numbers and she obtains 
different integers. On the other hand, Ben substracts each of his numbers from each of Ana’s numbers and he gets 
different integers.For each integer
, let 
be the number of ways that may be written as sum of one number of Ana and one number of Ben.a) Show that there exist an integer
such that,
b) Does there exist an integer such that,
Proposed by Besfort Shala, Kosovo
This post has been edited 3 times. Last edited by dangerousliri, Mar 8, 2022, 9:15 PM
,
are the initial sets of numbers of Ana and Ben.
as pigeons and 
as holes, by Pigeonhole there exists a 
.
