[/quote]
,
that is not isosceles at 
,
be its incircle, which is tangent to 
at 
,
and 
intersect the line passing through 
at 
and 
,
and perpendicular to 
intersect 
and 
at 
and 
,
are collinear and that 
are concurrent.
be a prime number. Prove that there exists an integer 
with 
such that neither 
nor 
is divisible by 
.
such that: 
be an odd prime number. Prove that there exist sequence 
Prove that
with 
,
denote the incenter and 
the midpoint of side 
.
perpendicular to 
intersects the tangent from 
> Show that 
.
![$P \in \mathbb R[x,y]$](http://latex.artofproblemsolving.com/8/a/0/8a0576299e976c51607959fb68c7c742b2a49c6a.png)
with ![$P \not\in \mathbb R[x] \cup \mathbb R[y]$](http://latex.artofproblemsolving.com/a/f/7/af71ba8c643d5bd8a2928fa6714ccd91713bd641.png)
and 
homeomorphismes of 
,
is an homoemorphisme too.
,
and 
are both integers.
be real numbers such that 
Prove that 
Let 
Prove that 
and 
Prove that
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.
