[/quote]
,
is an integer. Prove that 
are integers.
(
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
.
is the intersection of 
.
is the projection of 
.
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.
denote the smallest possible value of 
where 
are sets such that 
and 
whenever 
.
and 
intersect at 
of the two circles touches 
at 
respectively. Let 
be the symmetric points of 
intersects circles 
respectively which are distinct from 
and 
are equal.
Y by Rounak_iitr
Let 
and 
be points on the sides 
and 
of the triangle 
, respectively. The circumcircle of 
meets the circumcircle of 
for the second time at 
. The line 
meets the line 
at 
, while the line 
meets the line 
at 
. Prove that the lines 
, 
and 
pass through the same point.
Authored by Petar Filipovski
and 
be points on the sides 
and 
of the triangle 
, respectively. The circumcircle of 
meets the circumcircle of 
for the second time at 
. The line 
meets the line 
at 
, while the line 
meets the line 
at 
. Prove that the lines 
, 
and 
pass through the same point.Authored by Petar Filipovski
This post has been edited 2 times. Last edited by steppewolf, Feb 9, 2025, 10:35 PM
![[asy]
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[/asy]](http://latex.artofproblemsolving.com/4/d/e/4de3f2b4e05844d80f528f91b7b58bd6cd906027.png)
be intersections of 
as a circumcircle of 
, 
as a circumcircle of 
be intersections of the circumcircle of 
with respect to 
are concyclic ( 
as well)
done. 
(Similarly, 
)
(by 
and 
we get 
,
are isogonal w.r.t. 
.
are isogonal w.r.t. 
and 
,
are isogonal w.r.t. 
and 
,
and 
and 
,
are also isogonal w.r.t. 
and 
we get that 
,
and 
are totally not needed, the point 
and 
gives that 
,
and 
are collinear, done. 
