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.
be an acute triangle with incentre 
and 
. Let lines 
and 
intersect the circumcircle of at 
and 
, respectively. Consider points 
and 
such that 
and 
are parallelograms (with 
, and 
). Let 
be the point of intersection of lines 
and 
. Prove that points 
, and are concyclic.
Find the range of values that can be taken by :
are positive reals.
there are elements in an arithmetic progression in the order 
such that 
.
would be divisible by 
for any integer values of 
, provided 
is not a factor of . How to prove this?
be an integer. In a configuration of an 
board, each of the 
cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate 
counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of 
, the maximum number of good cells over all possible starting configurations.
types of candy, numerated type 1, type 2, ... type n. Initially she takes 
candy pieces and places them in a row on a table. Then, she chooses one of the following operations (if available) and executes it:
She eats a candy of type 
, and in its position in the row she places one candy type 
followed by one candy type 
(we consider type 
to be type 1, and type 0 to be type ).
She chooses two consecutive candies which are the same type, and eats them. for which Celeste can leave the table empty for any value of and any configuration of candies on the table.
such that 
has a prime divisor greater than 
.
be an interior point of the convex quadrilateral 
. Construct triangles 
and 
on the outside of the quadrilateral such that the similarities 
and 
hold. Let 
and be the projections of on the lines 
and 
, respectively. Prove that if the quadrilateral 
is cyclic, then![\[EF\cdot CD=EG\cdot DA=EH\cdot AB=EI\cdot BC.\]](http://latex.artofproblemsolving.com/0/8/e/08e13ecbf07fb1e8950fa34e40cb7f08a8c5a7e1.png)
+1 w
, let 
be all the positive integers smaller than that are coprime to . Find all 
such that
for all 
.
and 
are arranged so that 
lies on the segment 
and lies on the segment 
. Point 
is the midpoint of 
. Prove that 
.
of the triangle , meets its sides 
and 
at the points and respectively. The triangle 
is acute. The points 
are chosen on the side such that 
and 
. Prove that 
.
of the triangle , meets its sides and at the points and respectively. The triangle is acute. The points are chosen on the side such that and . Prove that .
and 
are cyclic so because 
is on the angle bisectors of 
and 
then 
and 
. Line from 
perpendicular to 
intersects at 
and lin from 
perpendicular to intersects at 
. By simple angle chasing we get that 
are collinear. Also 
and 
(because 
are the perpendicular bisectors of 
). From this we get that 
are the reflections of 
in so 
. Now:
which is equivalent to what we wanted.
and 
Prove that
be reals such that 
Prove that
by ASA, 
