[/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 David-Vieta
Let 
denotes the set of integers from 
to 
. A magician with an assistant perform a trick. The magician leaves the hall and the spectator writes a sequence of 
elements on the board from the set . The magician’s assistant looks at them and adds 
more elements from to the existing sequence. After that the spectator replaces three of these 
numbers by random elements of (it is permitted to change them by themselves, that is to not change anything at all, for example). The magician enters and looks at the resulting row of numbers and without error names the original numbers written by the spectator. Find the minimal possible for which the trick is possible.
denotes the set of integers from 
to 
. A magician with an assistant perform a trick. The magician leaves the hall and the spectator writes a sequence of 
elements on the board from the set . The magician’s assistant looks at them and adds 
more elements from to the existing sequence. After that the spectator replaces three of these 
numbers by random elements of (it is permitted to change them by themselves, that is to not change anything at all, for example). The magician enters and looks at the resulting row of numbers and without error names the original numbers written by the spectator. Find the minimal possible for which the trick is possible.
and assistant add 
.
are written in the black board. Spectator replaces at most three place of 
and let 
be the resulting sequence. Magician sees 
and he must guess original 
.
(See the popular notation) which means sending message with length 10 adding 
.
is very hard. At least we need some theory like Reed-Solomon error correction. ![$f\in Z_{239}[t]$](http://latex.artofproblemsolving.com/7/4/2/742c21f0eb1a3f16662f2f1c110500396d27d8fc.png)
of degree 
for which
at 
,
to the right-hand side. The spectator will change three numbers, the magician will comes in and tries all the supposed sets of 13 unchanged numbers. Each he will interpolate the corresponding points with a polynomial
.
