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 a convex pentagon such that 
. Suppose that the midpoint of 
is the circumcenter of triangle 
. Let 
be the circumcenter of triangle 
.
passes through the midpoint of segment 
.
such that for all 
the following holds:![\[f(x)+f(y)=f(xy)+f(f(x)+f(y))\]](http://latex.artofproblemsolving.com/a/c/a/aca098172ea7ce5663e9dd0dd10c2f1495630230.png)
be a cyclic pentagon inscribed in a circle of centre which has angles 
. Show that the diagonals 
and 
meet at a point belonging to the diameter .
and 
are the incenter and the 
-excenter of triangle 
, respectively.
be the midpoint of arc 
on the circumcircle, and 
be the foot of the to 
. The line 
intersects the circumcircle again at 
. For
on the circumcircle of triangle , let 
intersect at 
. Prove
are concyclic.
be an acute triangle. Points 
, and 
lie on a line in this order and satisfy 
. Let and 
be the midpoints of 
and 
, respectively. Suppose triangle 
is acute, and let 
be its orthocentre. Points 
and 
lie on lines 
and 
, respectively, such that 
and are concyclic and pairwise different, and 
and are concyclic and pairwise different. Prove that 
and are concyclic.
. 
are the midpoints of 
respectively. is the midpoint of arc , is symmetric to through . Let 
be the incenter and tangent of angles of triangle 
respectively. Prove that 
have a common point
be the circumcircle of triangle , is the midpoint of and 
be the second intersection of 
and . Tangent line of at intersects at , let 
be a transversal of and 
be intersections of 
with . Let 
be a tangent line of . Prove that 
is cyclic
be integers such that there are consecutive integers 
satisfy 
.
is a perfect square.
on one side and a on the other. Harry has 
of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly 
coins showing , then he turns over the 
th coin from the left; otherwise, all coins show and he stops. For example, if 
the process starting with the configuration 
would be 
, which stops after three operations., let 
be the number of operations before Harry stops. For example, 
and 
. Determine the average value of over all 
possible initial configurations .
can be connected by segments with points 
and 
. Is it possible to connect with broken lines, consisting of such segments the points
and 
?
and 
?
a
Prove that
