[/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 
.
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
In Tlaxcala, there is a transportation system that works through buses that travel from one city to another in one direction . A set 
of cities is said beautiful if it contains at least three different cities and from each city 
in at least two buses depart, each one goes directly to a different city in and none of them is (if there is a direct bus from to a city 
in , there is not necessarily a direct bus from to ). Show that if there exists a beautiful set of cities , then there exists a beautiful 
subset of , such that for any two cities in , you can get from one to another by taking buses that only pass through cities in .
Note: A bus goes directly from one city to another if it does not pass through any other city.
of cities is said beautiful if it contains at least three different cities and from each city 
in at least two buses depart, each one goes directly to a different city in and none of them is (if there is a direct bus from to a city 
in , there is not necessarily a direct bus from to ). Show that if there exists a beautiful set of cities , then there exists a beautiful 
subset of , such that for any two cities in , you can get from one to another by taking buses that only pass through cities in .Note: A bus goes directly from one city to another if it does not pass through any other city.
This post has been edited 1 time. Last edited by parmenides51, Nov 13, 2021, 11:20 PM
Reason: typo: removed <em>
Reason: typo: removed <em>
that we cannot get to from a city 
in 
This implies that the city 
that leads to 
that leads to 
and any city that 
Since there is always another city leading to any city leading to 
and any city that 
It's also a beautiful set, since each city in it leads to another two cities in turn. So if a set is not satisfactory, there exists a smaller beautiful subset of the set. Since the smallest possible set (of 3 cities) is satisfactory, for any 
