[/quote]
,
be an acute-angled scalene triangle with incircle 
and circumcircle 
.
at 
and the tangent to 
meets line 
.
again at 
and 
.
is a trapezium.
lines are drawn on the plane such that no three of them are concurrent and no two are parallel. At least 
of the bounded regions formed are colored black. A triangle formed by three lines is called a \textit{good triangle} if it lies entirely within a black region. Prove that there are at least 
good triangles. (A good triangle is a bounded region with finite area.)
be an arrangement of 
.
cups labeled 
,
-
Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most 
steps.
Prove that in at most 
steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
and 
for which there exists a natural numbe
is integer for all natural 
.
has 
.
be an arbitrary point on the nine point circle of 
intersects 
.
intersects 
at 
is the midpoint of 
.
that, for all 
,![\[\sum_{i=1}^n \frac{x_i^2}{\sum_{j=1}^i x_j}\geq C \ln n\left(\prod_{i=1}^n x_i\right)^{\frac{1}{n}}\]](http://latex.artofproblemsolving.com/4/8/b/48b9fc4af5a20de7fd9dfc45a74d5aa27e615e46.png)
that satisfy the condition![\[
\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 3.
\]](http://latex.artofproblemsolving.com/2/7/d/27d266570e3a105a4b416bb7c55904ffabf7a07c.png)
Find the maximum value of the expression![\[
P = \dfrac{yz}{\sqrt[3]{3y^2z^2+ 3x^2y^2z^2+ x^2z^2 + x^2y^2}}
+ \frac{xz}{\sqrt[3]{3x^2z^2 + 3x^2y^2z^2 + x^2y^2 + y^2z^2}}
+ \frac{xy}{\sqrt[3]{3x^2y^2 + 3x^2y^2z^2 +y^2z^2 + x^2z^2}}.
\]](http://latex.artofproblemsolving.com/b/a/f/bafc95e59d2f28dbb4e02771db38ce736e3fd45c.png)
.
.
-
numbers 
each equal to either 
or 
.
so that, for any 
and indices 
so that 
for all 
and 
such that
be reals such that 
and 
Find the value of 
Y by Rounak_iitr
Let 
be the number of positive integers with 
digits 
in base (where 
for all 
and 
) such that the polynomial
![\[d_9x^9+d_8x^8+\cdots+d_1x+d_0\]](//latex.artofproblemsolving.com/6/6/7/667e5a3d7f6c4f984e1278b5f76559c3b50c9690.png)
is irreducible in 
. Prove that 
is even.
(A polynomial is irreducible in if it cannot be factored into two non-constant polynomials with rational coefficients.)
be the number of positive integers with 
digits 
in base (where 
for all 
and 
) such that the polynomial![\[d_9x^9+d_8x^8+\cdots+d_1x+d_0\]](http://latex.artofproblemsolving.com/6/6/7/667e5a3d7f6c4f984e1278b5f76559c3b50c9690.png)
is irreducible in 
. Prove that 
is even.(A polynomial is irreducible in
if it cannot be factored into two non-constant polynomials with rational coefficients.)
then 
for all 
with 
as a polynomial is irreducible iff its reverse is.
else 
divides the polynomial, and 
works. This is because if 
was somehow not irreducible, i.e. 
for some nonconstant polynomials ![$g(x), h(x) \in \mathbb{Q}[x]$](http://latex.artofproblemsolving.com/7/b/5/7b596909c781d727c0920f07a722b1118e714905.png)
,
contradicting the fact that 
as 
,
.
,
is irreducible, then its reverse is also irreducible.
.
implies the polynomial is divisible by 
is palindromic, then 
multiplied by 
,
.
for which 
.
.
and vice-versa.
.
,
.
,
.
,
,
,
to get two polynomials. But if indeed 
,
is a factor, contradicting the irreducibility of 
,
.
,
.
and it can easily be checked that one is irreducible if and only if the other is
