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
a
Prove that
.
we have rectangle 
.
such that ![$[H\cap H_{x}]$](http://latex.artofproblemsolving.com/6/b/3/6b394579e1d80251a265f0701cc86646fdbb0643.png)
minimum, where ![$[S]$](http://latex.artofproblemsolving.com/6/0/0/600580b5de8084177c6fd276063e6f870b313abe.png)
is area of 
.
and 
(like in the diagram). Then we are looking for the minimum of ![$ [WXYZ]$](http://latex.artofproblemsolving.com/b/1/3/b132ea1d52066bd54b8dd717c14619f4e134cd40.png)
.![\begin{align*}[WXYZ]&=AB\cdot WX\\
&\ge AB\cdot A'B'\\
&= AB^2\end{align*}](http://latex.artofproblemsolving.com/9/a/5/9a577e9d4a83056c56056130c85f0fdcdc6ffc10.png)
is the perpendicular distance between the parallels 
and 
,
;
is a rotation of 
,
.
,
.
is between 
and 
which might need in that case we'll need that 
condition.