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.
where is defined below. Write your answer in interval notation.
be a positive integer. There are 
knights sitting at a round table. They consist of pairs of partners, each pair of which wishes to shake hands. A pair can shake hands only when next to each other. Every minute, one pair of adjacent knights swaps places. Find the minimum number of exchanges of adjacent knights such that, regardless of the initial arrangement, every knight can meet her partner and shake hands at some time.
be a cyclic quadrilateral, 
, 
, 
, 
, 
and 
Prove that:
, let nonnegative reals 
satisfy 
. Prove:
where 
. (The equality holds if 
and other 
being 0.)
+1 w
, where 
. This made me wonder, does there exist a nice family of solutions describing all of them?
be a triangle such that 
. Let the excircle opposite to A be tangent to the lines 
, and 
at points 
, and 
, respectively, and let 
be its centre. Let 
be a point on the side . The circumcircles of the triangles 
and 
intersect for the second time at 
. Let 
be the foot of the perpendicular from 
to the line 
. Prove that the points 
, and are collinear. opposite to is the circle that is tangent to the line segment , to the ray 
beyond 
, and to the ray 
beyond 
.)
be a convex pentagon with 
and 
. is located in the interior of the pentagon such that 
and 
. lies on the diagonal 
if and only if area 
+ area 
= area 
+ area 
.
be a triangle with incenter 
and circumcenter 
, such that 
. Let 
be the midpoint of , and let 
and be the points where 
intersects 
and , respectively. A line passing through and perpendicular to 
intersects 
and 
at 
and 
, respectively. Prove that the circle symmetric to the circumcircle of 
wrt 
is tangent to the circumcircle of 
.
be the set of all lines in the plane and let 
be a function that assigns to each line 
a point 
on 
. Suppose that for any point 
, and for any three lines 
passing through , the points 
, and lie on a circle. such that 
for any line passing through .
has 
, 
, 
, and 
. If 
is the intersection of 
and 
, what is the measure, in degrees, of 
?
for all 
.
such that 
such that
x>y>0
2
