AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
: 
such that:
for all x,y E 
denote the number of positive integer divisors of a positive integer 
(for example, 
). Given a polynomial 
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 . We then say the sequence has limit infinity if every integer occurs in this sequence only finitely many times (possibly not at all). for which the sequence 
, 
, . . . has limit infinity?
satisfying![\[f(x+f(y))=x+f(f(y))\]](http://latex.artofproblemsolving.com/8/d/a/8daa6bcb3155768ee1ba4bd22beb53a85a6e3923.png)
and 
, with the additional constraint 
.
be a positive number, and we show decimal part of the with 
.For a positive number with 
such that, 
=
.Find value of the
with Incenter 
and A-Excenter 
, let 
be the foot of perpendicular from 
to 
. Prove that bisects 
, 
, points 
are on different sides of line 
, and 
. Prove that 
are colinear.
cities in a country, where 
is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities and 
we define:
A 
between and as a sequence of distinct cities 
, 
, such that there are direct flights between 
and 
for every 
;
A 
between and as a path between and such that no other path between and has more cities;
A 
between and as a path between and such that no other path between and has fewer cities. and in the country, there exist a long path and a short path between them that have no cities in common (except and ). Let 
be the total number of pairs of cities in the country that are connected by direct flights. In terms of , find all possible values 
be a positive integer. Do there exist distinct -digit positive integers and , with each of their digits being 
or 
, such that 
divides 
?
and 
be the incircle and circumcircle of the acute triangle , respectively. Draw a square 
so that all of its sides are tangent to , and 
, 
are both on . Extend 
and 
, intersecting at 
and 
, respectively. Prove that 
and 
intersects on .
for which there exist positive integers , , and 
such that the number
has 
as its circumcenter, and 
as its orthocenter. The line 
and 
intersect the circumcircle of for the second time at points and 
, respectively. Let 
and 
be the circumcenters of triangle 
and 
respectively. If 
are not collinear, prove that 
intersects the midpoint of segment 
.
be a triangle with incircle 
, which touches sides 
and at points and , respectively. Choose points 
and 
on the line 
such that 
and 
. Let be the intersection of lines 
and 
. Define and 
as the intersections of 
and 
with lines 
and 
, respectively. Assume that 
is the intersection of 
and . Prove that 
.
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 th turn. On each of Poseidon’s turns, every cell which is adjacent to an already flooded cell and with no wall between them becomes flooded as well. Hephaestus wins if the levee forms a closed loop such that all flooded cells are contained in the interior of the loop — hence stopping the flood and saving the world. For which 
can Hephaestus guarantee victory in a finite number of turns no matter how Poseidon chooses the initial cells to flood?
, pairwise distinct except possibly 
, such that the set of walls is exactly 
. Once a wall is built it cannot be destroyed; in particular, if the levee is a closed loop (i.e. ) then Hephaestus cannot add more walls. Since each wall has length 
, the length of the levee is 
.
and are chosen inside the acute angle 
in such way that 
. Construct with ruler and compass the point at the side such that the bisector 
of triangle 
is perpendicular to 
.
and are chosen inside the acute angle in such way that . Construct with ruler and compass the point at the side such that the bisector of triangle is perpendicular to .
and 
Find the minimum 
