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.
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.
squares, we cover the area with several S-Tetrominos (=Z-Tetrominos) along the square so that in every square, there are two or fewer tiles covering that (tiles can be overlap). Find the maximum possible number of squares covered by at least one tile.
satisfies
for any non-negative integers 
and 
. Determine the number of possible sets of integers 
.
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle 
with circumcenter 
, centroid 
and orthocenter 
. Prove that 
is constant.
be the positive real number. Prove that
, such that for all positive reals 
, the following is true:
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.
, with two of the points, say 
in its interior.
meet two side of the triangle (recall that no three points are collinear), say and 
. Thus 
are in some order the vertices of a convex quadrilateral, and we are done.
points be , , 
, , and . We first consider , , , and . If each of these is in the exterior of the triangle determined by the other points, we are done by . If one of these is in the interior, however, WLOG let it be . We now proceed with casework based on the position of . is in the exterior of triangle 
, the lines , , and 
split the plane into regions. If is in the region determined by and , then 
is convex, and analogous reasoning applies to the other 
regions. is in the interior of triangle , extend bast to intersect 
at , and define and similarly. Then, if is in the interior of triangle 
or 
, 
or 
is convex respectively. Analogous reasoning applies to the other pairs of triangles. 
points forming a convex quadrilateral.
