[/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
For all integers 
, a non-negative integer 
is written on the point 
on the coordinate plane. Initially, 
and the value written on all remaining points is 
.
For integers
that satisfies 
, define 'Seehang' as the act of reducing 
by 
, selecting 3 of 
and increasing them by 1.
Prove that after a finite number of 'Seehang's, it cannot be
for all integers 
.
, a non-negative integer 
is written on the point 
on the coordinate plane. Initially, 
and the value written on all remaining points is 
.For integers
that satisfies 
, define 'Seehang' as the act of reducing 
by 
, selecting 3 of 
and increasing them by 1.Prove that after a finite number of 'Seehang's, it cannot be
for all integers 
.
counts the number of tokens on every point. Denote by 
the total number of points with at least one token on it.
.
after 
moves.
or 
makes a 3-move. Note that, due to 
token from a 'fuel point'. If it has more then one fuel point, then it can't make a 3-move. So the fuel point has given it a token at least twice, meaning one of it's moves is a 0 or 1-move. Pair these two. We now claim there's at most one overlap in this pairing.
after 
always, so there has to be some point with at least two tokens on it 
.
of original weight. Then the total weight of tokens is constant 4. If 
,
,
.
