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 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
be an integer. Consider an 
chessboard consisting of 
unit squares. A configuration of rooks on this board is peaceful if every row and every column contains exactly one rook. Find the greatest positive integer such that, for each peaceful configuration of rooks, there is a 
square which does not contain a rook on any of its 
unit squares.
students. Let, for 
, the 
student have 
many friends. For 
let us define 
to be the number of students who have strictly more than 
friends. Show that 
be a quadrilateral and let 
be a line. Let intersect the lines 
at points 
respectively. Given that these six points on are in the order 
, show that the circles with diameter 
are coaxal.
is 
. now cebisev.
is . now cebisev.
be the angles of a triangle opposite to its sides with lengths respectively. Prove the inequality![\[a\left(\frac{1}{\beta}+\frac{1}{\gamma}\right)+b\left(\frac{1}{\gamma}+\frac{1}{\alpha}\right)+c\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)\ge2\left(\frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}\right)\]](http://latex.artofproblemsolving.com/e/f/f/eff2334468c3af16d97f1d0ac22ef0b081ec6c1f.png)
a
Prove that
such that:
with all 
is real number
