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.
is the midpoint of the arc 
, containing 
, in the circumcircle of 
, and let 
be the -excircle's center. Assume that the external angle bisector of 
intersects at 
. Prove that 
is perpendicular to 
, where 
is the incenter of .
with 
and the circle 
of radius 
centered at 
Let 
be the midpoint of 
The line 
intersects a second time at 
Let 
be a point on such that 
Let 
be the intersection of 
and Prove that 
cities. For every two cities there exists exactly one direct one-way road between them. We say that a path from 
to 
is a sequence of roads such that one can move from to along this sequence without returning to an already visited city. A collection of paths is called diverse if no road belongs to two or more paths in the collection.
and 
be two distinct cities in Anisotropy. Let 
denote the maximal number of paths in a diverse collection of paths from to . Similarly, let 
denote the maximal number of paths in a diverse collection of paths from to . Prove that the equality 
holds if and only if the number of roads going out from is the same as the number of roads going out from . with incentre 
and excentres 
and 
, opposite the vertices 
and 
respectively. The incircle touches 
and 
at 
and 
respectively. Prove that the circles 
and 
have a common point other than .
and outside triangle such that 
and 
. Connect 
and 
, which intersect at point 
. Let 
intersect 
at point 
. Prove that 
.
such that for all integers 
and 
, ![\[ f(2x+f(y))+f(f(2x))=y. \]](http://latex.artofproblemsolving.com/b/0/d/b0d16e87a8b30ebec5997ed12254094cdc3125d5.png)
= 2∠AMP.
, and an odd prime 
not dividing any of them, and an integer such that if one considers the infinite sequence 
and looks at the highest power of that divides each of them, these powers are not all zero, and are all at most . Prove that there exists some 
(which may depend on 
) such that whenever divides an element of this sequence, the maximum power of that divides that element is exactly 
.
be a positive integer. Consider 
points on the circumference of a circle. pairs, and line segments are drawn connecting the points in each pair.
} such that for every 2 different number 
and 
in this set at last in one of the permutation comes exactly after
: 
such that:
convex polygons, determine the maximum possible number of edges in the obtained figure. polygons with 
vertices and 
edges in the resulting figure, then 
.)
is divided into 
equal right triangles and the square 
, as shown in the figure. Find the area of the square if 
.
,
and we want 
.
gives 
.