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.
+1 w
+2 w
be a set of integers with the following properties:
.
and 
, then 
.
, 
is composite, then all positive divisors of are in . contains all positive integers.
such that 
is an integer for every positive integer 
with circumcenter 
Let 
be an arbitrary point on 
such that is outside 
Let 
be an arbitrary point on 
cuts 
again at 
and 
cuts 
again at 
The intersection of 
and 
is 
Let 
and 
be the intersection of 
with 
and 
respectively such that , , are pairwise distinct.
, 
, 
are coaxial circles
, 
, 
, 
be 
integers. If 
is an integer, prove that the only solution to
is is 
.
be a cyclic quadrilateral an let be a point on the side 
The diagonals meets the segments 
at 
The line through parallel to 
mmets the extension of the side 
beyond 
at 
The line through parallel to 
meets the extension of the side beyond at 
Prove that the circumcircles of the triangles 
and 
are tangent .
such that 
, find the maximum possible number of integers 
such that 
such that 
is a perfect square but none of the integers 
are perfect squares.
and ![$P_1(x),P_2(x), \ldots P_n(x) \in \mathbb{Z}[x]$](http://latex.artofproblemsolving.com/c/6/4/c64176e52d94032e571b4579e35f9e6adfde675a.png)
such that 
. Prove that exists many 
such that every equation: 
has no real roots
, Its incircle 
and 
escircle 
are tangent to 
at 
and respectively. 
intersects line at 
. Line 
intersects at , and line 
intersects at 
. Line 
intersects at 
. Prove that 
.
be a positive integer. A subset 
with four distinct elements is special if there exists a rearrangement 
of 
such that 
. Prove that the set 
cannot be partitioned into special disjoint sets.
(first player) and 
, take alternate turns in playing a game using 2016 chips as follows: the player whose turn it is, must remove 
chips from the remaining pile of chips, where 
. No one can skip a turn. The player who at some point is unable to make a move (cannot remove chips from the pile) loses the game. Who among the two players can force a win on this game?
