[/quote]
,
,
,
and 
.
in the Cartesian plane such that all side lengths of 
and 
are both sides of 
b
and 
on 
and 
are on the same side of 
be midpoint of segment 
be midpoint of segment 
be intersection of 
with 
and 
,
such that 
(i.e: sum of powers of point 
![\[
\max(a_1, a_2) + \max(a_2, a_3), + \dots + \max(a_{n-1}, a_n) + \max(a_n, a_1),
\]](http://latex.artofproblemsolving.com/4/c/b/4cba741142419b031efdeba713748762fa6b2aac.png)
where 
runs over the set of permutations of 
.
,
denote the integers modulo ![$\mathbb{F}_p[x]$](http://latex.artofproblemsolving.com/e/5/a/e5a21492095c9116b100e2f7626a70cc26834e82.png)
be the set of polynomials with coefficients in ![$P(x) \in \mathbb{F}_p[x]$](http://latex.artofproblemsolving.com/7/a/a/7aa574df0669ce78536b1ecc5c949120aa2b3009.png)
such that for all integers 
,
such that 
.
quartic polynomials in 
![$f(x)\in\mathbb Z[x]$](http://latex.artofproblemsolving.com/2/9/5/2953d049559ec5d4020e6777d965bdfff9ebde0f.png)
have positive integer leading coefficient. Show that there exists infinte positive integer 
such that the number of digit that doesn'r equal to 
is no more than 
,
Determine 
.
has at least one pair of positive integer solution 
for each positive integer 
.
be an acute triangle. Denote by 
the foot of the perpendicular line drawn from the point 
to the side 
,
the orthocenter of 
be the point of intersection of the circumcircle 
of the triangle 
,
be the point of intersection (other than 
)
and the circle 
must hold.
the length of the line segment 
two not intersecting circles 
and 
are drawn such that 
at points 
respectively, and 
and 
at 
and 
respectively. It is known that 
.
and let 
be the foot of the
-
in line 
respectively such that 
while 
and 
and 
are tangent to each other.
Y by
Suppose 
die is laid down on an infinite grid of 
squares. The die starts perfectly aligned in one of the square of the grid with the 
facing up and the 
facing forward. The die can be "rolled" along an edge such that is moves exactly one unit up, down, left, or right and rotates 
correspondingly in that axis.
Determine, with proof, all possible squares relative to the starting square for which it is possible to roll the die to that square and have it in the same orientation ( facing up and facing forward).
(This problem may be hard to visualize/understand. If you don't understand the question, please ask and I can clarify.)
die is laid down on an infinite grid of 
squares. The die starts perfectly aligned in one of the square of the grid with the 
facing up and the 
facing forward. The die can be "rolled" along an edge such that is moves exactly one unit up, down, left, or right and rotates 
correspondingly in that axis.Determine, with proof, all possible squares relative to the starting square for which it is possible to roll the die to that square and have it in the same orientation (
facing up and facing forward).(This problem may be hard to visualize/understand. If you don't understand the question, please ask and I can clarify.)
