AoPS Academy
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, 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.
be an acute-angled triangle with circumcircle 
and circumcentre 
. Points 
and 
lie on such that 
and 
. Let 
meet 
at 
, and 
meet 
at 
.
and 
have an intersection lie on line 
.
be a positive integer. Alex plays on a row of 9 squares as follows. Initially, all squares are empty. In each turn, Alex must perform exactly one of the following moves:
Choose a number of the form 
, with 
a non-negative integer, and place it in an empty square.
Choose two (not necessarily consecutive) squares containing the same number, say . Replace the number in one of the squares with 
and erase the number in the other square.
, while the other squares are empty. Determine, as a function of , the maximum number of turns Alex can make.
be a convex hexagon with 
and 
.
which is equidistant from sides 
and 
.
and 
are the centers of mass of 
and 
, show that 
.
be a point in the plane, distinct from the vertices of 
. Consider 
the reflections of with respect to lines 
and 
, in this order.
are collinear if and only if lies on the circumcircle of . does not lie on the circumcircle of and the centroids of triangles and 
coincide, prove that is equilateral.
given that there exist positive integers 
satisfying
and , we define 
as follows:
where 
are all the positive divisors of . For example,
.
Find all positive integers such that 
.
Find all positive integers such that 
.
be an odd prime number. How many -element subsets 
of 
are there, the sum of whose elements is divisible by ?
be real numbers satisfying
be a fixed square and 
a variable point on segment 
. The square 
is constructed such that 
is on segment 
and 
is on segment 
. Let 
be the intersection point of lines 
and 
. Find the locus of as varies along segment .
be a positive integer. In an 
board, two opposite sides have been joined, forming a cylinder. Determine whether it is possible to place queens on the board such that no two threaten each other when: 
. 
.
denote the number of positive integers less than that are relatively prime to . Prove that there exists a positive integer for which the equation 
has at least 
solutions in .
that can be written in the form
where 
are pairwise coprime positive integers such that 
, 
, and 
.
is more clean, we redefine points. Let be a point on 
that 
, and be the orthocenter of . 
is the second intersection of and , and 
be the projection of to 
. 
is the intersection of 
and .
, we have that is the antipode of wrt . This implies that 
, so it suffices to prove that 
. From 
inversion, we can easily see that 
. This implies that
Therefore, 
are collinear.
is basically working backwards to the original problem lol
be the second intersection of with 
and 
the intersection of with . Denote 
the antipode of 
in 
and the intersection of and . Let 
be orthogonal projections from 
onto and 
the intersection of 
with 
.
are diametrically opposite in , thus 
. Same follows with 
in so 
. Thus we have symmetry between 
and 
w.r.t perpidencular bisector of .![\[ F(AG,OR) = (AG,OR) = C(AG,OR) = -1 \]](http://latex.artofproblemsolving.com/f/a/6/fa6f8984e7f0cd0b4512ff9ff74c7805357c672d.png)
Implying that 
bisects 
. In other words, 
is isoceles at 
, hence 
. Now, since 
is a parallelogram, it follows from symmetry that 
. All this gives us 
. With the above symmetrical relations, one can conclude that 
are collinear.
and![\[ f(x+y)+f(x^2+f(y))=f(f(x))^2+f(x)+f(y)+y,\forall x,y\in\mathbb{R}\]](http://latex.artofproblemsolving.com/c/8/9/c895ae7fdf8d7f284ac9fc94cc077d6edad6cbf0.png)
.
.
is the projection of 
passes through the orthocenter of 