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.
. Initially, the numbers 
are written on a board. In each move, two numbers are selected such that their difference is also present on the board. This difference is then erased from the board. (For example, if the numbers 
and 
are on the board, then 
can be erased as 
, or 
as 
, or 
as 
.) is it possible to end with only one number remaining on the board?
. Points 
and 
lie on segments 
and 
, respectively. Lines 
and 
intersect at point 
. The circumcircles of triangles 
and 
intersect at a second point 
. The circumcircles of triangles and 
intersect at a second point 
. Point 
lies on segment 
such that 
. Prove that triangles 
and 
are similar.
, where 
and 
.
satisfies the following property:![\[
f\left( \lambda x + (1 - \lambda)x',\; \lambda y + (1 - \lambda)y' \right) > \min\{f(x, y),\; f(x', y')\}
\]](http://latex.artofproblemsolving.com/3/b/d/3bd9fbf82b0b1d98a3c68745bb3f159cd9e2529a.png)
and for all 
.
be a real numbers such that 
Then find maximum value of 
segments and 
are parallel. Angle bisectors of 
and 
meet at . Angle bisectors of 
and 
meet at . Prove that 
is cyclic
such that 
is even , 
is odd and 
is divisible by 
.Prove that there exists a prime number 
such that is divisible by 
and 
, then find the maximum value of ![$\sqrt{ad^3 g^4} +\sqrt[3]{be^3 h^4} + \sqrt[8]{cf^3 k^4}$](http://latex.artofproblemsolving.com/b/f/a/bfa17e8bb4cc05a84ed6a757f506d6d0ed2b0081.png)
such that there exists a prime number , such that 
is a power of . is a number of the form 
where is a positive integer.
be an acute triangle with orthocenter 
and circumcenter 
. Let and be points on the line segments and respectively such that 
is a parallelogram. Prove that 
.
, let 
be the midpoint of side 
. Let the line perpendicular to at point 
intersect 
at 
. If 
is tangent to at 
, find 
.
, the incircle 
touches at . Let 
be the excenter of opposite to 
, and let be the midpoint of 
. Prove that the circumcircle of triangle 
is tangent to .
, the incircle touches at . Let be the excenter of opposite to , and let be the midpoint of . Prove that the circumcircle of triangle is tangent to .
be the intersection of and .
is tangent to . By homothety sending to 
centered at , we get that if 
, then 
, where is the centre of . Also if is the midpoint of , we know that 
is the perpendicular bisector of , hence we get that 
is the midline of 
, where is touch point of incircle with and 
is touch point of -excircle with . We conclude that 
is the midpoint of 
.
Prove that
and 
Prove that
and 
.
,
is tangent to 
be touch points of incircle with 
be the antipode of 
concur at say 
and 
,
bisects 
.
.
,
.
,
are concyclic, as desired.