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.
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)
.
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.
and be positive integers. For a connected simple graph 
on vertices and edges, we consider the number 
of orientations of (all of) its edges so that, in the resulting directed graph, every vertex has even outdegree. only depends on and , and determine its value.
such that![\[(a^2 + b^2 + c^2)^2 = 3(a^3b + b^3c + c^3a) + 1. \]](http://latex.artofproblemsolving.com/9/e/4/9e400fd807a9535ede983b75dcc307d8b75a35b6.png)
so that 
is an integer.
, 
and be midpoints of sides 
and , respectively, and let be circumcenter of acute-angled triangle . Circumcircles of triangles 
and 
intersect at two different points 
and inside of triangle . Prove that ![\[\angle BAX=\angle CAY.\]](http://latex.artofproblemsolving.com/3/9/9/39910fdd000cbed820b33ac08a8200ca06ed8193.png)
is a positive integer. Call all positive divisors of which are different from 
and beautiful divisors.We call a special number when it has at least 
beautiful divisors and difference of any beautiful divisors divides as well. Find all special numbers.
with an apex angle equal to 
. Point is located inside the triangle so that
and 
. Find the value of angle 
.
Just construct as circumcenter of 
And observe 
Hence 
is also isosceles So 
Hence 
and 
.
