[/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
There are 6 squares in a row. Each one is labeled with the name of Ana or Beto and with a number from 1 to 6, using each number without repetition. Ana and Beto take turns painting each square according to the order of the numbers on the labels. Whoever paints the square will be the person whose name is on the label. When painting, the person can choose to paint the square either red or blue. Beto wins if at the end there are the same number of blue squares as red squares, and Ana wins otherwise. In how many of all the possible ways of labeling the squares can Beto ensure his victory?
The following is an example of a labeling of the labels.
![[asy]
size(12cm);
draw((0,0)--(6,0)--(6,-1)--(0,-1)--cycle);
for (int i=1; i<6; ++i) {
draw((i,0)--(i,-1));
}
for (int i=1; i<6; ++i) {
draw((i,0)--(i,-1.25));
}
draw((0,0)--(6,0)--(6,-1.25)--(0,-1.25)--cycle);
for (int i=1; i<7; ++i) {
draw((i-0.5,-1)--(i-0.5,-1.25));
}
label("Ana", (0.25, -1.125));
label("Beto", (1.25, -1.125));
label("Ana", (2.25, -1.125));
label("Beto", (3.25, -1.125));
label("Ana", (4.25, -1.125));
label("Beto", (5.25, -1.125));
label("1", (0.75, -1.125));
label("3", (1.75, -1.125));
label("5", (2.75, -1.125));
label("2", (3.75, -1.125));
label("4", (4.75, -1.125));
label("6", (5.75, -1.125));
[/asy]](//latex.artofproblemsolving.com/e/e/5/ee5df72b487bc558ac8c6868671fd5aa5ee7e412.png)
First Ana paints the first square, then Beto paints the fourth square, then Beto paints the second square, and so on.
The following is an example of a labeling of the labels.
![[asy]
size(12cm);
draw((0,0)--(6,0)--(6,-1)--(0,-1)--cycle);
for (int i=1; i<6; ++i) {
draw((i,0)--(i,-1));
}
for (int i=1; i<6; ++i) {
draw((i,0)--(i,-1.25));
}
draw((0,0)--(6,0)--(6,-1.25)--(0,-1.25)--cycle);
for (int i=1; i<7; ++i) {
draw((i-0.5,-1)--(i-0.5,-1.25));
}
label("Ana", (0.25, -1.125));
label("Beto", (1.25, -1.125));
label("Ana", (2.25, -1.125));
label("Beto", (3.25, -1.125));
label("Ana", (4.25, -1.125));
label("Beto", (5.25, -1.125));
label("1", (0.75, -1.125));
label("3", (1.75, -1.125));
label("5", (2.75, -1.125));
label("2", (3.75, -1.125));
label("4", (4.75, -1.125));
label("6", (5.75, -1.125));
[/asy]](http://latex.artofproblemsolving.com/e/e/5/ee5df72b487bc558ac8c6868671fd5aa5ee7e412.png)
First Ana paints the first square, then Beto paints the fourth square, then Beto paints the second square, and so on.
