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.
+2 w
, such that for all positive reals 
, the following is true:
+1 w
and 
are the incenter and the 
-excenter of triangle 
, respectively.
be the midpoint of arc 
on the circumcircle, and 
be the foot of the to 
. The line 
intersects the circumcircle again at 
. For
on the circumcircle of triangle , let 
intersect at 
. Prove
are concyclic.
grid. Alice goes first, and they take turns drawing a black point from the coordinate set![\[\{(i, j) \mid i, j \in \mathbb{N}, 1 \leq i, j \leq n\}\]](http://latex.artofproblemsolving.com/4/3/f/43f7493d1dff43079eb5481ee17bc8f16d7b9553.png)
There is a constraint that the distance between any two black points cannot be an integer. The player who cannot draw a black point loses. Find all integers 
such that Alice has a winning strategy. grid, where 
. Prove that the remainder can be covered by copies of the figures of 
or 
unit squares depicted in the drawing below.
for which there exists a positive integer 
with the property that is divisible by and 
is divisible by 
.
denote the set of real numbers. Find all functions 
such that![\[f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)\]](http://latex.artofproblemsolving.com/5/b/e/5be34cb26d276bae3c60fb299ad66d83bc158a01.png)
.
+1 w
be a function that satisfies
for all positive integer pairs 
Find all possible functions 
as the set of all positive integers.) be a triangle with orthocenter 
and circumcenter 
. Let 
be a point in the plane such that 
. Let 
and 
be the reflections of in the lines 
and 
, respectively. Let be the orthogonal projection of onto . Let 
be the orthogonal projection of onto . Assume that 
and 
. Prove that 
.
satisfying
for all integers .
be a cyclic quadrilateral. A circle centered at passes through 
and and meets lines 
and again at points 
and 
(distinct from 
). Let denote the orthocenter of triangle 
Prove that if lines 
are concurrent, then triangle and 
are similar.
