[/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 ys-lg, lkjin7932
For a competition a school wants to nominate a team of 
students, where is a given positive integer. Each member of the team has to compete in the three disciplines juggling, singing and mental arithmetic. To qualify for the team, the 
students of the school compete in qualifying competitions, determining a unique ranking in each of the three disciplines. The school now wants to nominate a team satisfying the following condition:
If a student 
is not nominated for the team, there is a student 
on the team who defeated in at least two disciplines.
Determine all positive integers such that for any combination of rankings, a team can be chosen to satisfy the condition , when
a)
,
b)
.
students, where is a given positive integer. Each member of the team has to compete in the three disciplines juggling, singing and mental arithmetic. To qualify for the team, the 
students of the school compete in qualifying competitions, determining a unique ranking in each of the three disciplines. The school now wants to nominate a team satisfying the following condition:
If a student 
is not nominated for the team, there is a student 
on the team who defeated in at least two disciplines.Determine all positive integers
such that for any combination of rankings, a team can be chosen to satisfy the condition , whena)
,b)
.
is 
t
t
students in grade 
)
is at least as strong as 
.
or am I missing something?)
satisfies the condition.
.
and 
.
,
or the singing ability not weaker than 
is 
.
,
.
'
'
.
can defeat all students whose juggling ability is not weaker than 
.
satisfies the condition as required.
