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 
.
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.
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 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 . be a positive integer. Find the number of permutations 
, 
, 
of the
, 
, 
, satisfying
.
such that
non-negative real numbers,but 
0.Prove that![\[1\leq \frac{a+b}{a+4b+c}+\frac{b+c}{b+4c+a}+\frac{c+a}{c+4a+b}\leq \frac{4}{3}\]](http://latex.artofproblemsolving.com/8/b/7/8b7d7b8cbcab8f2eed31e17aac717fa3c70c92f1.png)
are positive real numbers, then
with 
. Let point be orthocenter 
and midpoint . It is also known that 
perpendicular to . Let be a point on the segment 
such that 
and point be the intersection of 
and 
. Prove that 
concyclic
denote the incircle and three excircles against 
Let 
be the mixtilinear incircles against 
and 
their corresponding mixtilinear excircles. 
and 
are tangent to rays 
through 
and respectively. It's well-known that lines 
and 
pass through the incenter and A-excenter 
respectively. Thus, from the similar kites 
it follows that
a
Prove that
.
be the ratio of the radii of the circles tangent to 
,
and 
.
and 
are defined analogously. Prove that 
.