[/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
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.
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 GeoKing
In a given group of people 
, each member has at least two acquaintances from . Moreover, for each cycle 
in (here '
' means that 
and 
are acquaintances), each 
knows exactly two other members 
of the cycle. Prove that there exist 
such that each of them has exactly two acquaintances in , and 
have at least one common acquaintance in the group.
Authored by Mirko Petrusevski
, each member has at least two acquaintances from . Moreover, for each cycle 
in (here '
' means that 
and 
are acquaintances), each 
knows exactly two other members 
of the cycle. Prove that there exist 
such that each of them has exactly two acquaintances in , and 
have at least one common acquaintance in the group.Authored by Mirko Petrusevski
This post has been edited 2 times. Last edited by steppewolf, Feb 9, 2025, 10:35 PM
Anyway the proof should be correct now.
.
.
,
,
denote the unique path from 
to 
in 
,
denote the other endpoint of the back-edge at 
be the first vertex after 
.
be the cycle consisting of the path in 
that is a descendent of 
is closer to the root than 
is a chord in the cycle 
,
,
,
ends at 
is further away from the root than 
.
is a chord in the cycle 
,
