[/quote]
,
:
such that:
for all x,y E 
denote the number of positive integer divisors of a positive integer 
(for example, 
)
with integer coefficients, we define a sequence 
of nonnegative integers by setting![\[a_n =\begin{cases}\gcd(P(n), \tau (P(n)))&\text{if }P(n) > 0\\0 &\text{if }P(n) \leq0\end{cases}\]](http://latex.artofproblemsolving.com/7/5/9/759f4e6c8511d920b858c4234eff7a53f38ac289.png)
for each positive integer 
,
,
satisfying![\[f(x+f(y))=x+f(f(y))\]](http://latex.artofproblemsolving.com/8/d/a/8daa6bcb3155768ee1ba4bd22beb53a85a6e3923.png)
and 
,
.
be a positive number, and we show decimal part of the 
.
such that, 
=
.
with Incenter 
and A-Excenter 
,
be the foot of perpendicular from 
to 
,
are on different sides of line 
.
are colinear.
is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities 
we define:
A 
between 
,
,
and 
for every 
;
A 
between 
A 
between 
be the total number of pairs of cities in the country that are connected by direct flights. In terms of 
or 
,
divides 
?
and 
Find the minimum 
and 
be the incircle and circumcircle of the acute triangle 
so that all of its sides are tangent to 
and 
,
and 
,
and 
intersects on 
for which there exist positive integers 
such that the number
as its circumcenter, and 
as its orthocenter. The line 
and 
intersect the circumcircle of 
,
and 
be the circumcenters of triangle 
and 
respectively. If 
are not collinear, prove that 
intersects the midpoint of segment 
.
,
and 
and 
on the line 
such that 
and 
.
and 
.
as the intersections of 
and 
with lines 
and 
,
is the intersection of 
and 
.
be a real number. Hephaestus and Poseidon play a turn-based game on an infinite grid of unit squares. Before the game starts, Poseidon chooses a finite number of cells to be flooded. Hephaestus is building a levee, which is a subset of unit edges of the grid (called walls) forming a connected, non-self-intersecting path or loop*.
after his 
can Hephaestus guarantee victory in a finite number of turns no matter how Poseidon chooses the initial cells to flood?
,
,
.
,
.
Y by Adventure10, Mango247
Let 
and 
denote the angles of the triangle 
. The perpendicular bisector of 
intersects 
at the point 
, the perpendicular bisector of 
intersects it at 
. Prove that 
implies 
(or in other words: Prove that a sufficient condition for 
is ). Show that this condition is not necessary, and give a necessary and sufficient condition for .
and 
denote the angles of the triangle 
. The perpendicular bisector of 
intersects 
at the point 
, the perpendicular bisector of 
intersects it at 
. Prove that 
implies 
(or in other words: Prove that a sufficient condition for 
is ). Show that this condition is not necessary, and give a necessary and sufficient condition for .
.
then the necessary and sufficient condition is that which was already stated 
)
being in the set 
.
