[/quote]
,
are defined by 
and ![\[a_{n + 1} = \frac {1 + a_n + a_nb_n}{b_n}, \quad b_{n + 1} = \frac {1 + b_n + a_nb_n}{a_n}.\]](http://latex.artofproblemsolving.com/e/4/9/e497e9872753206cebdf99b9a830e82821eb9bc3.png)
.
be a rational number and let 
be a positive integer. Prove that the polynomial 
is irreducible in the ring ![$\mathbb{Q}[X]$](http://latex.artofproblemsolving.com/f/3/d/f3dc4b61f66a97c235268da5643f5bebbc69c9bd.png)
of polynomials with rational coefficients.
for positive integers 
if they come in the top 
across all ratings for which they are eligible.
be a parabola that passes through the vertices of a triangle 
.
that is tangent to 
such that![$$\left(a+b+c+d-K\right)^2+2\left[1-\left(3\sqrt2-4\right)K\right]abcd\geq\left(3\sqrt2-K\right)^2$$](http://latex.artofproblemsolving.com/3/5/3/35382fd52c29efa5185d1b9fbeef944a0fd2fee3.png)
for all 
satisfying
.
be its circumcircle, 
its orthocenter, and 
the foot of the altitude from 
.
be the midpoint of 
.
be the point on 
such that 
and let 
.
,
,
and 
are tangent to each other.
be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:![\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]](http://latex.artofproblemsolving.com/5/5/3/553527fc40a1b7d29dbdfb4b5e63635308f46c85.png)
and 
.
, for all 
are different and chosen from the set 
.![\[\frac{(V+U+Q+A+R)^2}{V-U-Q+A+R}=V^{{{U^Q}^A}^R}.\]](http://latex.artofproblemsolving.com/d/4/f/d4fd4785add9eec6ad20ad288503712c8e31686a.png)
and 
,
intersects 
,
lie on 
be the circumcentre of triangle 
.
meets 
and meets 
.
.
.
.
Y by
Let"s have a 3xn matrice with the following properties:
The firs row of the matrice is 1,2,3,... ,n in this order.
The second and the third rows are permutations of the first.
Very important, that in each column thera are different entries.
How many matrices with thees properties are there?
The answer for 2xn matrices is well-known, but what is the answer for 3xn, or for kxn ( k<=n) ?
The firs row of the matrice is 1,2,3,... ,n in this order.
The second and the third rows are permutations of the first.
Very important, that in each column thera are different entries.
How many matrices with thees properties are there?
The answer for 2xn matrices is well-known, but what is the answer for 3xn, or for kxn ( k<=n) ?
