AoPS Academy
be a parallelogram with 
A point 
is chosen on the extension of ray 
past 
The circumcircle of 
meets the segment 
again at 
The circumcircle of triangle 
meets the segment 
at 
Prove that lines 
are concurrent.
be an isosceles triangle (
) with incenter 
. Circle 
passes through 
and and is tangent to 
. intersects 
and circumcircle of at 
and 
, respectively. Let 
be the midpoint of and 
be the midpoint of 
. Prove that 
, 
and 
are concurrent.
be a convex quadrilateral and let the diagonals and 
intersect at 
. Let 
be respectively the incentres of triangles 
. Let 
be respectively the excentres of triangles opposite . Show that lie on a circle if and only if lie on a circle.
be an acute triangle with orthocenter 
, and let 
be a point on the side , lying strictly between 
and . The points and are the feet of the altitudes from and , respectively. Denote by 
is the circumcircle of 
, and let 
be the point on such that 
is a diameter of . Analogously, denote by 
the circumcircle of triangle 
, and let 
be the point such that 
is a diameter of . Prove that 
and are collinear.
such that if any 
elements are removed from the set 
, one can still find distinct numbers among the remaining elements with sum .
from each of the integers at two neighboring vertices and adding 
to the opposite vertex, which is not adjacent to either of the first two vertices. (The amount and the vertices chosen can vary from turn to turn.) The game is won at a certain vertex if, after some number of turns, that vertex has the number 2011 and the other four vertices have the number 0. Prove that for any choice of the initial integers, there is exactly one vertex at which the game can be won.
index this yr (118.5 on amc10+ 9 on AIME) i’m in 7th rn btw first year comp math also. I will grind so hard until like 30 hrs/week. I’m ok at proofs.
is selected inside acute 
so that 
and 
. Point 
is chosen on ray so that 
. Let be the midpoint of . is tangent to the circumcircle of triangle 
.
that satisfy![\[
f(x^2-y)+2yf(x)=f(f(x))+f(y)
\]](http://latex.artofproblemsolving.com/f/b/b/fbb693770cd90d8a144026608074122e54da5802.png)
for all 
.
and 
be positive integers. Let 
be the set of integer points 
with 
and 
. A configuration of 
rectangles is called happy if each point in is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd.
and different odd prime numbers 
with
(mod 
), 
(mod 
), ..., 
(mod 
).
for some 
be divisible by 
for exactly one 
of 
?
so 
which is a contradiction.
, 
, 
. Furthermore we can also create infinitely many examples with any with the help of Dirichlet theorem on primes and of generators
